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Properties

Label 2-31e2-31.2-c1-0-54
Degree 22
Conductor 961961
Sign −0.997+0.0753i-0.997 + 0.0753i
Analytic cond. 7.673627.67362
Root an. cond. 2.770132.77013
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.335 − 0.243i)2-s + (0.335 − 0.243i)3-s + (−0.565 − 1.73i)4-s + 5-s − 0.171·6-s + (−0.127 − 0.393i)7-s + (−0.490 + 1.50i)8-s + (−0.874 + 2.68i)9-s + (−0.335 − 0.243i)10-s + (−1.00 − 3.08i)11-s + (−0.612 − 0.445i)12-s + (−3.09 + 2.25i)13-s + (−0.0530 + 0.163i)14-s + (0.335 − 0.243i)15-s + (−2.42 + 1.76i)16-s + (1.80 − 5.54i)17-s + ⋯
L(s)  = 1  + (−0.236 − 0.172i)2-s + (0.193 − 0.140i)3-s + (−0.282 − 0.869i)4-s + 0.447·5-s − 0.0700·6-s + (−0.0483 − 0.148i)7-s + (−0.173 + 0.533i)8-s + (−0.291 + 0.896i)9-s + (−0.105 − 0.0769i)10-s + (−0.302 − 0.929i)11-s + (−0.176 − 0.128i)12-s + (−0.859 + 0.624i)13-s + (−0.0141 + 0.0436i)14-s + (0.0865 − 0.0628i)15-s + (−0.606 + 0.440i)16-s + (0.436 − 1.34i)17-s + ⋯

Functional equation

Λ(s)=(961s/2 ΓC(s) L(s)=((−0.997+0.0753i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0753i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(961s/2 ΓC(s+1/2) L(s)=((−0.997+0.0753i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0753i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 961961    =    31231^{2}
Sign: −0.997+0.0753i-0.997 + 0.0753i
Analytic conductor: 7.673627.67362
Root analytic conductor: 2.770132.77013
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ961(374,⋅)\chi_{961} (374, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 961, ( :1/2), −0.997+0.0753i)(2,\ 961,\ (\ :1/2),\ -0.997 + 0.0753i)

Particular Values

L(1)L(1) ≈\approx 0.0233267−0.617873i0.0233267 - 0.617873i
L(12)L(\frac12) ≈\approx 0.0233267−0.617873i0.0233267 - 0.617873i
L(32)L(\frac{3}{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)
bad31 1 1
good2 1+(0.335+0.243i)T+(0.618+1.90i)T2 1 + (0.335 + 0.243i)T + (0.618 + 1.90i)T^{2}
3 1+(−0.335+0.243i)T+(0.927−2.85i)T2 1 + (-0.335 + 0.243i)T + (0.927 - 2.85i)T^{2}
5 1−T+5T2 1 - T + 5T^{2}
7 1+(0.127+0.393i)T+(−5.66+4.11i)T2 1 + (0.127 + 0.393i)T + (-5.66 + 4.11i)T^{2}
11 1+(1.00+3.08i)T+(−8.89+6.46i)T2 1 + (1.00 + 3.08i)T + (-8.89 + 6.46i)T^{2}
13 1+(3.09−2.25i)T+(4.01−12.3i)T2 1 + (3.09 - 2.25i)T + (4.01 - 12.3i)T^{2}
17 1+(−1.80+5.54i)T+(−13.7−9.99i)T2 1 + (-1.80 + 5.54i)T + (-13.7 - 9.99i)T^{2}
19 1+(3.57+2.59i)T+(5.87+18.0i)T2 1 + (3.57 + 2.59i)T + (5.87 + 18.0i)T^{2}
23 1+(−1.23+3.80i)T+(−18.6−13.5i)T2 1 + (-1.23 + 3.80i)T + (-18.6 - 13.5i)T^{2}
29 1+(5.52+4.01i)T+(8.96+27.5i)T2 1 + (5.52 + 4.01i)T + (8.96 + 27.5i)T^{2}
37 1+T+37T2 1 + T + 37T^{2}
41 1+(−6.05−4.39i)T+(12.6+38.9i)T2 1 + (-6.05 - 4.39i)T + (12.6 + 38.9i)T^{2}
43 1+(8.81+6.40i)T+(13.2+40.8i)T2 1 + (8.81 + 6.40i)T + (13.2 + 40.8i)T^{2}
47 1+(7.81−5.67i)T+(14.5−44.6i)T2 1 + (7.81 - 5.67i)T + (14.5 - 44.6i)T^{2}
53 1+(1.80−5.54i)T+(−42.8−31.1i)T2 1 + (1.80 - 5.54i)T + (-42.8 - 31.1i)T^{2}
59 1+(3.29−2.39i)T+(18.2−56.1i)T2 1 + (3.29 - 2.39i)T + (18.2 - 56.1i)T^{2}
61 1−2.82T+61T2 1 - 2.82T + 61T^{2}
67 1−3.24T+67T2 1 - 3.24T + 67T^{2}
71 1+(−0.0219+0.0675i)T+(−57.4−41.7i)T2 1 + (-0.0219 + 0.0675i)T + (-57.4 - 41.7i)T^{2}
73 1+(−0.565−1.73i)T+(−59.0+42.9i)T2 1 + (-0.565 - 1.73i)T + (-59.0 + 42.9i)T^{2}
79 1+(−2.08+6.42i)T+(−63.9−46.4i)T2 1 + (-2.08 + 6.42i)T + (-63.9 - 46.4i)T^{2}
83 1+(8.14+5.91i)T+(25.6+78.9i)T2 1 + (8.14 + 5.91i)T + (25.6 + 78.9i)T^{2}
89 1+(1.38+4.26i)T+(−72.0+52.3i)T2 1 + (1.38 + 4.26i)T + (-72.0 + 52.3i)T^{2}
97 1+(−1.59−4.91i)T+(−78.4+57.0i)T2 1 + (-1.59 - 4.91i)T + (-78.4 + 57.0i)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

−9.624473897633268696458034876125, −8.955637560229606784277821570906, −8.077353363318389909777278706162, −7.10253411403187890065378400369, −6.05514154414205557248586571350, −5.25443760442636536626899706409, −4.50497547103155691378959651773, −2.77818011815767627582475851594, −1.95791126517290227916977239341, −0.28218391925381551714308506094, 1.97528934845772330902290957294, 3.26912413271597384975892766400, 4.02364988197992490867794547603, 5.25747623449343855498335853548, 6.22592049219369445286279461447, 7.22484788100807740332813307316, 7.997849639512335563190360846203, 8.714623845014306132443106274695, 9.721216737444878659724818637144, 9.963093475571762571167269443385

Graph of the ZZ-function along the critical line