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Properties

Label 2-976-61.52-c1-0-29
Degree $2$
Conductor $976$
Sign $-0.493 + 0.869i$
Analytic cond. $7.79339$
Root an. cond. $2.79166$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.89 + 1.37i)3-s + (−0.701 − 2.15i)5-s + (−2.88 − 3.96i)7-s + (0.768 + 2.36i)9-s + 0.886i·11-s − 4.51·13-s + (1.64 − 5.05i)15-s + (−5.01 + 1.62i)17-s + (−2.88 − 2.09i)19-s − 11.4i·21-s + (0.910 − 0.295i)23-s + (−0.126 + 0.0921i)25-s + (0.370 − 1.14i)27-s + 0.133i·29-s + (1.85 + 2.54i)31-s + ⋯
L(s)  = 1  + (1.09 + 0.794i)3-s + (−0.313 − 0.965i)5-s + (−1.08 − 1.49i)7-s + (0.256 + 0.788i)9-s + 0.267i·11-s − 1.25·13-s + (0.424 − 1.30i)15-s + (−1.21 + 0.395i)17-s + (−0.661 − 0.480i)19-s − 2.50i·21-s + (0.189 − 0.0616i)23-s + (−0.0253 + 0.0184i)25-s + (0.0713 − 0.219i)27-s + 0.0247i·29-s + (0.332 + 0.457i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.493 + 0.869i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 976 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.493 + 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(976\)    =    \(2^{4} \cdot 61\)
Sign: $-0.493 + 0.869i$
Analytic conductor: \(7.79339\)
Root analytic conductor: \(2.79166\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{976} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 976,\ (\ :1/2),\ -0.493 + 0.869i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.517968 - 0.889441i\)
\(L(\frac12)\) \(\approx\) \(0.517968 - 0.889441i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
61 \( 1 + (-6.44 + 4.41i)T \)
good3 \( 1 + (-1.89 - 1.37i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (0.701 + 2.15i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (2.88 + 3.96i)T + (-2.16 + 6.65i)T^{2} \)
11 \( 1 - 0.886iT - 11T^{2} \)
13 \( 1 + 4.51T + 13T^{2} \)
17 \( 1 + (5.01 - 1.62i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (2.88 + 2.09i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-0.910 + 0.295i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 - 0.133iT - 29T^{2} \)
31 \( 1 + (-1.85 - 2.54i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (4.33 + 5.97i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-3.13 + 2.27i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + (-1.59 - 0.517i)T + (34.7 + 25.2i)T^{2} \)
47 \( 1 + 4.07T + 47T^{2} \)
53 \( 1 + (-0.335 - 0.108i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-0.654 + 0.900i)T + (-18.2 - 56.1i)T^{2} \)
67 \( 1 + (-6.70 + 2.17i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (-14.8 - 4.80i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-1.11 + 3.42i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (1.13 + 0.370i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (1.81 + 1.31i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-8.54 + 11.7i)T + (-27.5 - 84.6i)T^{2} \)
97 \( 1 + (-2.66 + 1.93i)T + (29.9 - 92.2i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.640891231797355784807627248543, −9.011051888246612669821166568992, −8.270355507383278135072025776577, −7.26836305874161398117971454269, −6.57328341070469766573683277447, −4.85501806173224643690162687700, −4.28741714305223005047397733234, −3.56288970309555158025012332720, −2.37124264019279088856913811073, −0.37708526742675421871597868880, 2.33893240490889323631661160676, 2.60814355511728498440138474734, 3.56755930451802301134686225848, 5.14791721862997062994909533614, 6.45162635791045678596752867656, 6.81378665660757435058699758184, 7.80835016685990289524362798317, 8.622554091767726291262081541160, 9.278966211348543234571853010584, 10.05266174822841670381079958250

Graph of the $Z$-function along the critical line