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Properties

Label 2-31e2-31.15-c0-0-0
Degree 22
Conductor 961961
Sign 0.800−0.599i0.800 - 0.599i
Analytic cond. 0.4796010.479601
Root an. cond. 0.6925320.692532
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  + (0.809 − 0.587i)2-s − 5-s + (−0.309 + 0.951i)7-s + (0.309 + 0.951i)8-s + (0.309 + 0.951i)9-s + (−0.809 + 0.587i)10-s + (0.309 + 0.951i)14-s + (0.809 + 0.587i)16-s + (0.809 + 0.587i)18-s + (0.809 − 0.587i)19-s + (0.309 − 0.951i)35-s + (0.309 − 0.951i)38-s + (−0.309 − 0.951i)40-s + (0.809 − 0.587i)41-s + (−0.309 − 0.951i)45-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)2-s − 5-s + (−0.309 + 0.951i)7-s + (0.309 + 0.951i)8-s + (0.309 + 0.951i)9-s + (−0.809 + 0.587i)10-s + (0.309 + 0.951i)14-s + (0.809 + 0.587i)16-s + (0.809 + 0.587i)18-s + (0.809 − 0.587i)19-s + (0.309 − 0.951i)35-s + (0.309 − 0.951i)38-s + (−0.309 − 0.951i)40-s + (0.809 − 0.587i)41-s + (−0.309 − 0.951i)45-s + ⋯

Functional equation

Λ(s)=(961s/2 ΓC(s) L(s)=((0.800−0.599i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 - 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(961s/2 ΓC(s) L(s)=((0.800−0.599i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 - 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 961961    =    31231^{2}
Sign: 0.800−0.599i0.800 - 0.599i
Analytic conductor: 0.4796010.479601
Root analytic conductor: 0.6925320.692532
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ961(573,⋅)\chi_{961} (573, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 961, ( :0), 0.800−0.599i)(2,\ 961,\ (\ :0),\ 0.800 - 0.599i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.2183001491.218300149
L(12)L(\frac12) ≈\approx 1.2183001491.218300149
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)
bad31 1 1
good2 1+(−0.809+0.587i)T+(0.309−0.951i)T2 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2}
3 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
5 1+T+T2 1 + T + T^{2}
7 1+(0.309−0.951i)T+(−0.809−0.587i)T2 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2}
11 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
13 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
17 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
19 1+(−0.809+0.587i)T+(0.309−0.951i)T2 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2}
23 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
29 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
37 1−T2 1 - T^{2}
41 1+(−0.809+0.587i)T+(0.309−0.951i)T2 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2}
43 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
47 1+(1.61+1.17i)T+(0.309+0.951i)T2 1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2}
53 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
59 1+(−0.809−0.587i)T+(0.309+0.951i)T2 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2}
61 1−T2 1 - T^{2}
67 1−2T+T2 1 - 2T + T^{2}
71 1+(0.309+0.951i)T+(−0.809+0.587i)T2 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2}
73 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
79 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
83 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
89 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
97 1+(0.309−0.951i)T+(−0.809−0.587i)T2 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)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

−10.62321659388754841866965603933, −9.520881951687328372734467396747, −8.500118979968597322051741501239, −7.892758070131895069074756497659, −7.03021885067196271822906544245, −5.61640950188835646776086368368, −4.92070586494791923869689276324, −3.99992463075727683850497885626, −3.07408450536182380060690743409, −2.12279685446713081119735335241, 0.990534801976033516665361488151, 3.41600072498515458727613160522, 3.91402161414175759583179288958, 4.74563035446808683017939455733, 5.90119211391729129107191851526, 6.74687668537547120398363177976, 7.36202380002155801657416394872, 8.162678412961214099815384512034, 9.545013209075178272523631630261, 9.972325190319962556273292130807

Graph of the ZZ-function along the critical line