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Properties

Label 1-297-297.200-r0-0-0
Degree 11
Conductor 297297
Sign 0.299−0.954i0.299 - 0.954i
Analytic cond. 1.379261.37926
Root an. cond. 1.379261.37926
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.438 − 0.898i)2-s + (−0.615 − 0.788i)4-s + (0.997 + 0.0697i)5-s + (−0.0348 + 0.999i)7-s + (−0.978 + 0.207i)8-s + (0.5 − 0.866i)10-s + (0.719 − 0.694i)13-s + (0.882 + 0.469i)14-s + (−0.241 + 0.970i)16-s + (−0.104 − 0.994i)17-s + (0.978 − 0.207i)19-s + (−0.559 − 0.829i)20-s + (0.939 − 0.342i)23-s + (0.990 + 0.139i)25-s + (−0.309 − 0.951i)26-s + ⋯
L(s)  = 1  + (0.438 − 0.898i)2-s + (−0.615 − 0.788i)4-s + (0.997 + 0.0697i)5-s + (−0.0348 + 0.999i)7-s + (−0.978 + 0.207i)8-s + (0.5 − 0.866i)10-s + (0.719 − 0.694i)13-s + (0.882 + 0.469i)14-s + (−0.241 + 0.970i)16-s + (−0.104 − 0.994i)17-s + (0.978 − 0.207i)19-s + (−0.559 − 0.829i)20-s + (0.939 − 0.342i)23-s + (0.990 + 0.139i)25-s + (−0.309 − 0.951i)26-s + ⋯

Functional equation

Λ(s)=(297s/2 ΓR(s) L(s)=((0.299−0.954i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.299 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(297s/2 ΓR(s) L(s)=((0.299−0.954i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.299 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 297297    =    33⋅113^{3} \cdot 11
Sign: 0.299−0.954i0.299 - 0.954i
Analytic conductor: 1.379261.37926
Root analytic conductor: 1.379261.37926
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ297(200,⋅)\chi_{297} (200, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 297, (0: ), 0.299−0.954i)(1,\ 297,\ (0:\ ),\ 0.299 - 0.954i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.404527172−1.031596089i1.404527172 - 1.031596089i
L(12)L(\frac12) ≈\approx 1.404527172−1.031596089i1.404527172 - 1.031596089i
L(1)L(1) ≈\approx 1.292779712−0.6443199199i1.292779712 - 0.6443199199i
L(1)L(1) ≈\approx 1.292779712−0.6443199199i1.292779712 - 0.6443199199i

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
11 1 1
good2 1+(0.438−0.898i)T 1 + (0.438 - 0.898i)T
5 1+(0.997+0.0697i)T 1 + (0.997 + 0.0697i)T
7 1+(−0.0348+0.999i)T 1 + (-0.0348 + 0.999i)T
13 1+(0.719−0.694i)T 1 + (0.719 - 0.694i)T
17 1+(−0.104−0.994i)T 1 + (-0.104 - 0.994i)T
19 1+(0.978−0.207i)T 1 + (0.978 - 0.207i)T
23 1+(0.939−0.342i)T 1 + (0.939 - 0.342i)T
29 1+(−0.882+0.469i)T 1 + (-0.882 + 0.469i)T
31 1+(0.961+0.275i)T 1 + (0.961 + 0.275i)T
37 1+(−0.978−0.207i)T 1 + (-0.978 - 0.207i)T
41 1+(−0.882−0.469i)T 1 + (-0.882 - 0.469i)T
43 1+(−0.766+0.642i)T 1 + (-0.766 + 0.642i)T
47 1+(0.615−0.788i)T 1 + (0.615 - 0.788i)T
53 1+(0.809+0.587i)T 1 + (0.809 + 0.587i)T
59 1+(0.374−0.927i)T 1 + (0.374 - 0.927i)T
61 1+(−0.961+0.275i)T 1 + (-0.961 + 0.275i)T
67 1+(0.173−0.984i)T 1 + (0.173 - 0.984i)T
71 1+(0.104+0.994i)T 1 + (0.104 + 0.994i)T
73 1+(−0.669+0.743i)T 1 + (-0.669 + 0.743i)T
79 1+(−0.438+0.898i)T 1 + (-0.438 + 0.898i)T
83 1+(−0.719−0.694i)T 1 + (-0.719 - 0.694i)T
89 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
97 1+(−0.997+0.0697i)T 1 + (-0.997 + 0.0697i)T
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   L(s)=∏p (1−αp p−s)−1L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−25.638232048285094470202873117617, −24.6097100891218613750215356775, −23.89079109649722090696822277543, −22.995046188338837900714076840819, −22.1584220960854496645884268504, −21.15367443421065987502799495674, −20.58088489823186163419902704749, −19.05351547556389238656507420759, −18.01892928714355420993773417313, −17.08934513654327840314593731497, −16.68022316255002883048719711391, −15.48778689941938777365271501858, −14.42937215311432251862820112088, −13.53715449380780178434025547108, −13.221025371408769446500018799421, −11.79084613461161507743047806408, −10.46723916029688932507132805876, −9.44390856308633985164940805859, −8.45662798265228823780723932119, −7.239679410050579366192473851955, −6.411197119588251285109180317724, −5.46285395288498208666982786550, −4.31463193527452718863632950722, −3.26094274599546926700751653736, −1.45614762242950874673891094400, 1.26134056654454969141545415057, 2.51526720876764470313530530127, 3.28593872489525045527563746834, 5.0796732943432136998538634251, 5.54403129187255697737579686477, 6.74552266195831832969460972301, 8.62966329182713257554747216320, 9.33919071620117747342542858493, 10.27947179420189165000449253591, 11.28466514811109544324604993023, 12.23662699943480037887264845209, 13.21352662444693770301163704599, 13.870188022139380651270202112441, 14.9422903744145244951133498777, 15.842702032987540869749524176657, 17.34551058761012307257895078890, 18.360463317018568972814650361582, 18.63563246160129511939117219823, 20.10182150966512939617901181079, 20.8120017860996335130707046615, 21.588487217351474255464005801338, 22.41667889412787230075649680363, 23.00989551277378193834082729010, 24.519414820613054147537727379514, 24.95471072347500763950349175852

Graph of the ZZ-function along the critical line