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Properties

Label 2-968-88.53-c1-0-81
Degree $2$
Conductor $968$
Sign $-0.943 + 0.330i$
Analytic cond. $7.72951$
Root an. cond. $2.78020$
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  + (−0.126 + 1.40i)2-s + (−1.10 − 1.52i)3-s + (−1.96 − 0.357i)4-s + (0.468 − 0.152i)5-s + (2.28 − 1.36i)6-s + (0.141 + 0.102i)7-s + (0.752 − 2.72i)8-s + (−0.165 + 0.508i)9-s + (0.155 + 0.679i)10-s + (1.63 + 3.38i)12-s + (−5.24 − 1.70i)13-s + (−0.162 + 0.186i)14-s + (−0.749 − 0.544i)15-s + (3.74 + 1.40i)16-s + (0.209 + 0.645i)17-s + (−0.694 − 0.296i)18-s + ⋯
L(s)  = 1  + (−0.0896 + 0.995i)2-s + (−0.637 − 0.878i)3-s + (−0.983 − 0.178i)4-s + (0.209 − 0.0681i)5-s + (0.931 − 0.556i)6-s + (0.0534 + 0.0388i)7-s + (0.266 − 0.963i)8-s + (−0.0550 + 0.169i)9-s + (0.0490 + 0.214i)10-s + (0.470 + 0.977i)12-s + (−1.45 − 0.473i)13-s + (−0.0434 + 0.0497i)14-s + (−0.193 − 0.140i)15-s + (0.936 + 0.351i)16-s + (0.0508 + 0.156i)17-s + (−0.163 − 0.0699i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 + 0.330i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.943 + 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $-0.943 + 0.330i$
Analytic conductor: \(7.72951\)
Root analytic conductor: \(2.78020\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{968} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 968,\ (\ :1/2),\ -0.943 + 0.330i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0129392 - 0.0760675i\)
\(L(\frac12)\) \(\approx\) \(0.0129392 - 0.0760675i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.126 - 1.40i)T \)
11 \( 1 \)
good3 \( 1 + (1.10 + 1.52i)T + (-0.927 + 2.85i)T^{2} \)
5 \( 1 + (-0.468 + 0.152i)T + (4.04 - 2.93i)T^{2} \)
7 \( 1 + (-0.141 - 0.102i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (5.24 + 1.70i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.209 - 0.645i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.257 + 0.353i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 - 6.88T + 23T^{2} \)
29 \( 1 + (-0.829 + 1.14i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.29 - 3.97i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (3.83 - 5.27i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (7.91 - 5.75i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 8.22iT - 43T^{2} \)
47 \( 1 + (6.27 - 4.55i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (9.43 + 3.06i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-5.55 + 7.65i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (9.13 - 2.96i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 - 5.24iT - 67T^{2} \)
71 \( 1 + (-1.42 - 4.39i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (5.54 + 4.02i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-2.09 + 6.44i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (6.92 - 2.25i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + 9.61T + 89T^{2} \)
97 \( 1 + (-1.03 + 3.19i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.587145000833794110368748790068, −8.615875470144235494064602680163, −7.68633301226799425409840262586, −7.01553956070545587930404276293, −6.42772805997627121034355100907, −5.36112903506028770839389320576, −4.86872239936215907830253333045, −3.29955383869721981986833110388, −1.52352559668601506917384030108, −0.04039506508125898362760160086, 1.89862227439507811441640555890, 3.06493511669006572131544980317, 4.31114715462897188987417629590, 4.88179849375085652770343614094, 5.68144224616576252211765805270, 7.07575199951864399686342103938, 8.058689457945127389240593448526, 9.266717482808656330442634392536, 9.672831063801275937027532375696, 10.44237451408522271684134201883

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