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Properties

Label 2-34e2-1156.135-c0-0-0
Degree 22
Conductor 11561156
Sign 0.960+0.278i0.960 + 0.278i
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  + (0.982 + 0.183i)2-s + (0.932 + 0.361i)4-s + (−0.554 − 0.895i)5-s + (0.850 + 0.526i)8-s + (0.602 − 0.798i)9-s + (−0.380 − 0.981i)10-s + (−1.02 − 0.634i)13-s + (0.739 + 0.673i)16-s + (0.739 + 0.673i)17-s + (0.739 − 0.673i)18-s + (−0.193 − 1.03i)20-s + (−0.0483 + 0.0971i)25-s + (−0.890 − 0.811i)26-s + (−0.576 + 1.48i)29-s + (0.602 + 0.798i)32-s + ⋯
L(s)  = 1  + (0.982 + 0.183i)2-s + (0.932 + 0.361i)4-s + (−0.554 − 0.895i)5-s + (0.850 + 0.526i)8-s + (0.602 − 0.798i)9-s + (−0.380 − 0.981i)10-s + (−1.02 − 0.634i)13-s + (0.739 + 0.673i)16-s + (0.739 + 0.673i)17-s + (0.739 − 0.673i)18-s + (−0.193 − 1.03i)20-s + (−0.0483 + 0.0971i)25-s + (−0.890 − 0.811i)26-s + (−0.576 + 1.48i)29-s + (0.602 + 0.798i)32-s + ⋯

Functional equation

Λ(s)=(1156s/2 ΓC(s) L(s)=((0.960+0.278i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1156s/2 ΓC(s) L(s)=((0.960+0.278i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11561156    =    22⋅1722^{2} \cdot 17^{2}
Sign: 0.960+0.278i0.960 + 0.278i
Analytic conductor: 0.5769190.576919
Root analytic conductor: 0.7595510.759551
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1156(135,⋅)\chi_{1156} (135, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1156, ( :0), 0.960+0.278i)(2,\ 1156,\ (\ :0),\ 0.960 + 0.278i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.7949716071.794971607
L(12)L(\frac12) ≈\approx 1.7949716071.794971607
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+(−0.982−0.183i)T 1 + (-0.982 - 0.183i)T
17 1+(−0.739−0.673i)T 1 + (-0.739 - 0.673i)T
good3 1+(−0.602+0.798i)T2 1 + (-0.602 + 0.798i)T^{2}
5 1+(0.554+0.895i)T+(−0.445+0.895i)T2 1 + (0.554 + 0.895i)T + (-0.445 + 0.895i)T^{2}
7 1+(−0.273−0.961i)T2 1 + (-0.273 - 0.961i)T^{2}
11 1+(0.739+0.673i)T2 1 + (0.739 + 0.673i)T^{2}
13 1+(1.02+0.634i)T+(0.445+0.895i)T2 1 + (1.02 + 0.634i)T + (0.445 + 0.895i)T^{2}
19 1+(−0.932+0.361i)T2 1 + (-0.932 + 0.361i)T^{2}
23 1+(−0.273−0.961i)T2 1 + (-0.273 - 0.961i)T^{2}
29 1+(0.576−1.48i)T+(−0.739−0.673i)T2 1 + (0.576 - 1.48i)T + (-0.739 - 0.673i)T^{2}
31 1+(0.445+0.895i)T2 1 + (0.445 + 0.895i)T^{2}
37 1+(0.365+0.0339i)T+(0.982+0.183i)T2 1 + (0.365 + 0.0339i)T + (0.982 + 0.183i)T^{2}
41 1+(0.942−0.469i)T+(0.602−0.798i)T2 1 + (0.942 - 0.469i)T + (0.602 - 0.798i)T^{2}
43 1+(−0.0922+0.995i)T2 1 + (-0.0922 + 0.995i)T^{2}
47 1+(0.273−0.961i)T2 1 + (0.273 - 0.961i)T^{2}
53 1+(−0.329+0.436i)T+(−0.273−0.961i)T2 1 + (-0.329 + 0.436i)T + (-0.273 - 0.961i)T^{2}
59 1+(0.850+0.526i)T2 1 + (0.850 + 0.526i)T^{2}
61 1+(1.72−0.489i)T+(0.850−0.526i)T2 1 + (1.72 - 0.489i)T + (0.850 - 0.526i)T^{2}
67 1+(−0.932+0.361i)T2 1 + (-0.932 + 0.361i)T^{2}
71 1+(−0.273−0.961i)T2 1 + (-0.273 - 0.961i)T^{2}
73 1+(−1.07+1.17i)T+(−0.0922−0.995i)T2 1 + (-1.07 + 1.17i)T + (-0.0922 - 0.995i)T^{2}
79 1+(0.932−0.361i)T2 1 + (0.932 - 0.361i)T^{2}
83 1+(0.602+0.798i)T2 1 + (0.602 + 0.798i)T^{2}
89 1+(1.67−1.03i)T+(0.445−0.895i)T2 1 + (1.67 - 1.03i)T + (0.445 - 0.895i)T^{2}
97 1+(0.576+0.435i)T+(0.273+0.961i)T2 1 + (0.576 + 0.435i)T + (0.273 + 0.961i)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

−10.06706295507933575298841500907, −9.054358931473362507320195822207, −8.089829924418128477321066536574, −7.42326737145714415271847980229, −6.56716115062235479634095840296, −5.49955892845792557878241895539, −4.80192745306717756972990534308, −3.92454956943859865693968939851, −3.06433417513601149814297403798, −1.45058883492430517636768617904, 1.95925444809955159535992324438, 2.90576493237275265068458125636, 3.93015085882034604302827768668, 4.77396694661198314396230551195, 5.63595430999851957833209812169, 6.87315530382093505474111182115, 7.26682839154861595135534226752, 7.997854580358814393410188040471, 9.579716829864106219208320699949, 10.19488023814903711866728805278

Graph of the ZZ-function along the critical line