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Properties

Label 2-34e2-1156.735-c0-0-0
Degree 22
Conductor 11561156
Sign −0.624−0.781i-0.624 - 0.781i
Analytic cond. 0.5769190.576919
Root an. cond. 0.7595510.759551
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.798 + 0.602i)2-s + (0.273 + 0.961i)4-s + (−0.739 + 1.67i)5-s + (−0.361 + 0.932i)8-s + (0.995 − 0.0922i)9-s + (−1.59 + 0.890i)10-s + (−1.85 − 0.719i)13-s + (−0.850 + 0.526i)16-s + (0.850 − 0.526i)17-s + (0.850 + 0.526i)18-s + (−1.81 − 0.252i)20-s + (−1.58 − 1.73i)25-s + (−1.04 − 1.69i)26-s + (1.11 + 0.621i)29-s + (−0.995 − 0.0922i)32-s + ⋯
L(s)  = 1  + (0.798 + 0.602i)2-s + (0.273 + 0.961i)4-s + (−0.739 + 1.67i)5-s + (−0.361 + 0.932i)8-s + (0.995 − 0.0922i)9-s + (−1.59 + 0.890i)10-s + (−1.85 − 0.719i)13-s + (−0.850 + 0.526i)16-s + (0.850 − 0.526i)17-s + (0.850 + 0.526i)18-s + (−1.81 − 0.252i)20-s + (−1.58 − 1.73i)25-s + (−1.04 − 1.69i)26-s + (1.11 + 0.621i)29-s + (−0.995 − 0.0922i)32-s + ⋯

Functional equation

Λ(s)=(1156s/2 ΓC(s) L(s)=((−0.624−0.781i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.624 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1156s/2 ΓC(s) L(s)=((−0.624−0.781i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.624 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11561156    =    22⋅1722^{2} \cdot 17^{2}
Sign: −0.624−0.781i-0.624 - 0.781i
Analytic conductor: 0.5769190.576919
Root analytic conductor: 0.7595510.759551
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1156(735,⋅)\chi_{1156} (735, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1156, ( :0), −0.624−0.781i)(2,\ 1156,\ (\ :0),\ -0.624 - 0.781i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.4029655931.402965593
L(12)L(\frac12) ≈\approx 1.4029655931.402965593
L(1)L(1) 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)
bad2 1+(−0.798−0.602i)T 1 + (-0.798 - 0.602i)T
17 1+(−0.850+0.526i)T 1 + (-0.850 + 0.526i)T
good3 1+(−0.995+0.0922i)T2 1 + (-0.995 + 0.0922i)T^{2}
5 1+(0.739−1.67i)T+(−0.673−0.739i)T2 1 + (0.739 - 1.67i)T + (-0.673 - 0.739i)T^{2}
7 1+(0.183+0.982i)T2 1 + (0.183 + 0.982i)T^{2}
11 1+(0.526+0.850i)T2 1 + (0.526 + 0.850i)T^{2}
13 1+(1.85+0.719i)T+(0.739+0.673i)T2 1 + (1.85 + 0.719i)T + (0.739 + 0.673i)T^{2}
19 1+(−0.273+0.961i)T2 1 + (-0.273 + 0.961i)T^{2}
23 1+(0.183+0.982i)T2 1 + (0.183 + 0.982i)T^{2}
29 1+(−1.11−0.621i)T+(0.526+0.850i)T2 1 + (-1.11 - 0.621i)T + (0.526 + 0.850i)T^{2}
31 1+(−0.673+0.739i)T2 1 + (-0.673 + 0.739i)T^{2}
37 1+(−1.87−0.629i)T+(0.798+0.602i)T2 1 + (-1.87 - 0.629i)T + (0.798 + 0.602i)T^{2}
41 1+(−0.0373−0.806i)T+(−0.995+0.0922i)T2 1 + (-0.0373 - 0.806i)T + (-0.995 + 0.0922i)T^{2}
43 1+(0.445+0.895i)T2 1 + (0.445 + 0.895i)T^{2}
47 1+(0.982+0.183i)T2 1 + (0.982 + 0.183i)T^{2}
53 1+(0.365−0.0339i)T+(0.982−0.183i)T2 1 + (0.365 - 0.0339i)T + (0.982 - 0.183i)T^{2}
59 1+(0.932+0.361i)T2 1 + (0.932 + 0.361i)T^{2}
61 1+(−0.0762−0.0521i)T+(0.361+0.932i)T2 1 + (-0.0762 - 0.0521i)T + (0.361 + 0.932i)T^{2}
67 1+(0.273−0.961i)T2 1 + (0.273 - 0.961i)T^{2}
71 1+(−0.183−0.982i)T2 1 + (-0.183 - 0.982i)T^{2}
73 1+(0.352−1.49i)T+(−0.895−0.445i)T2 1 + (0.352 - 1.49i)T + (-0.895 - 0.445i)T^{2}
79 1+(0.961+0.273i)T2 1 + (0.961 + 0.273i)T^{2}
83 1+(0.0922−0.995i)T2 1 + (0.0922 - 0.995i)T^{2}
89 1+(−1.48+0.576i)T+(0.739−0.673i)T2 1 + (-1.48 + 0.576i)T + (0.739 - 0.673i)T^{2}
97 1+(0.748+0.621i)T+(0.183+0.982i)T2 1 + (0.748 + 0.621i)T + (0.183 + 0.982i)T^{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

−10.21935499940888942025973771617, −9.762593977150344847138138418787, −8.096228351774222616595616232390, −7.52257914605923706546861241910, −7.04309841285125537248997669176, −6.31276947355514285771174942542, −5.08259477077727639052495317671, −4.26149583129492170160781036040, −3.15422621956112565266401166439, −2.61054270770460349266109043411, 1.04121157962032752093557134549, 2.25235674811074659467623072174, 3.80775828342829547855740950117, 4.60560529532195168929351880555, 4.90828957427837386733795194153, 6.08058663990796720446572543242, 7.33087650887909444780417822621, 7.929799132200622901843788299782, 9.229669862935233361047810481994, 9.650610946266755469408015830769

Graph of the ZZ-function along the critical line