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Properties

Label 2-13e2-13.12-c7-0-23
Degree 22
Conductor 169169
Sign −0.832+0.554i-0.832 + 0.554i
Analytic cond. 52.793052.7930
Root an. cond. 7.265887.26588
Motivic weight 77
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6.78i·2-s − 30.1·3-s + 81.9·4-s + 93.2i·5-s − 204. i·6-s + 1.61e3i·7-s + 1.42e3i·8-s − 1.27e3·9-s − 632.·10-s + 6.35e3i·11-s − 2.47e3·12-s − 1.09e4·14-s − 2.81e3i·15-s + 821.·16-s + 8.37e3·17-s − 8.66e3i·18-s + ⋯
L(s)  = 1  + 0.599i·2-s − 0.645·3-s + 0.640·4-s + 0.333i·5-s − 0.387i·6-s + 1.78i·7-s + 0.983i·8-s − 0.583·9-s − 0.200·10-s + 1.44i·11-s − 0.413·12-s − 1.06·14-s − 0.215i·15-s + 0.0501·16-s + 0.413·17-s − 0.350i·18-s + ⋯

Functional equation

Λ(s)=(169s/2 ΓC(s) L(s)=((−0.832+0.554i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(169s/2 ΓC(s+7/2) L(s)=((−0.832+0.554i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 169169    =    13213^{2}
Sign: −0.832+0.554i-0.832 + 0.554i
Analytic conductor: 52.793052.7930
Root analytic conductor: 7.265887.26588
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: χ169(168,⋅)\chi_{169} (168, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 169, ( :7/2), −0.832+0.554i)(2,\ 169,\ (\ :7/2),\ -0.832 + 0.554i)

Particular Values

L(4)L(4) ≈\approx 1.5650991291.565099129
L(12)L(\frac12) ≈\approx 1.5650991291.565099129
L(92)L(\frac{9}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad13 1 1
good2 1−6.78iT−128T2 1 - 6.78iT - 128T^{2}
3 1+30.1T+2.18e3T2 1 + 30.1T + 2.18e3T^{2}
5 1−93.2iT−7.81e4T2 1 - 93.2iT - 7.81e4T^{2}
7 1−1.61e3iT−8.23e5T2 1 - 1.61e3iT - 8.23e5T^{2}
11 1−6.35e3iT−1.94e7T2 1 - 6.35e3iT - 1.94e7T^{2}
17 1−8.37e3T+4.10e8T2 1 - 8.37e3T + 4.10e8T^{2}
19 1−2.12e4iT−8.93e8T2 1 - 2.12e4iT - 8.93e8T^{2}
23 1−1.36e3T+3.40e9T2 1 - 1.36e3T + 3.40e9T^{2}
29 1+9.65e4T+1.72e10T2 1 + 9.65e4T + 1.72e10T^{2}
31 1+1.11e5iT−2.75e10T2 1 + 1.11e5iT - 2.75e10T^{2}
37 1−4.65e5iT−9.49e10T2 1 - 4.65e5iT - 9.49e10T^{2}
41 1+9.74e4iT−1.94e11T2 1 + 9.74e4iT - 1.94e11T^{2}
43 1−4.03e5T+2.71e11T2 1 - 4.03e5T + 2.71e11T^{2}
47 1+1.90e4iT−5.06e11T2 1 + 1.90e4iT - 5.06e11T^{2}
53 1−1.14e6T+1.17e12T2 1 - 1.14e6T + 1.17e12T^{2}
59 1+2.81e6iT−2.48e12T2 1 + 2.81e6iT - 2.48e12T^{2}
61 1+5.46e5T+3.14e12T2 1 + 5.46e5T + 3.14e12T^{2}
67 1−1.93e6iT−6.06e12T2 1 - 1.93e6iT - 6.06e12T^{2}
71 1+1.12e6iT−9.09e12T2 1 + 1.12e6iT - 9.09e12T^{2}
73 1+3.91e4iT−1.10e13T2 1 + 3.91e4iT - 1.10e13T^{2}
79 1+2.19e6T+1.92e13T2 1 + 2.19e6T + 1.92e13T^{2}
83 1+9.73e6iT−2.71e13T2 1 + 9.73e6iT - 2.71e13T^{2}
89 1−8.59e6iT−4.42e13T2 1 - 8.59e6iT - 4.42e13T^{2}
97 1+7.31e6iT−8.07e13T2 1 + 7.31e6iT - 8.07e13T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−11.97558650771254353487797902676, −11.33281795419901500655195091822, −10.11573651131084291337104961009, −8.879873090343055450456757339501, −7.81055022307665857541736038132, −6.62548380647639100092500312500, −5.81028820835913716747886893152, −5.04750310732258932502984352325, −2.85810668782116824202375271308, −1.90544191033579816910801408773, 0.49292601228208192088554720595, 1.05979560112169512093227330578, 2.95611514605203479414626367568, 4.00218408797948299848612686339, 5.51908899970035118205344888297, 6.63852934129408032233767666269, 7.58434356857282800871632835310, 8.963612571143143223417522450689, 10.45794744626678238270158398336, 10.88399882792408293811121637233

Graph of the ZZ-function along the critical line