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Properties

Label 2-968-88.51-c1-0-31
Degree $2$
Conductor $968$
Sign $-0.860 - 0.508i$
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  + (0.468 + 1.33i)2-s + (0.127 − 0.393i)3-s + (−1.56 + 1.25i)4-s + (1.55 + 2.14i)5-s + (0.585 − 0.0138i)6-s + (1.01 + 3.12i)7-s + (−2.40 − 1.49i)8-s + (2.28 + 1.66i)9-s + (−2.12 + 3.07i)10-s + (0.292 + 0.774i)12-s + (1.10 + 0.801i)13-s + (−3.69 + 2.82i)14-s + (1.04 − 0.338i)15-s + (0.871 − 3.90i)16-s + (−3.22 − 4.44i)17-s + (−1.14 + 3.83i)18-s + ⋯
L(s)  = 1  + (0.331 + 0.943i)2-s + (0.0739 − 0.227i)3-s + (−0.780 + 0.625i)4-s + (0.695 + 0.957i)5-s + (0.239 − 0.00564i)6-s + (0.384 + 1.18i)7-s + (−0.848 − 0.529i)8-s + (0.762 + 0.554i)9-s + (−0.672 + 0.973i)10-s + (0.0845 + 0.223i)12-s + (0.305 + 0.222i)13-s + (−0.988 + 0.754i)14-s + (0.269 − 0.0874i)15-s + (0.217 − 0.975i)16-s + (−0.782 − 1.07i)17-s + (−0.270 + 0.903i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.860 - 0.508i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.860 - 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $-0.860 - 0.508i$
Analytic conductor: \(7.72951\)
Root analytic conductor: \(2.78020\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{968} (403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 968,\ (\ :1/2),\ -0.860 - 0.508i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.549030 + 2.00825i\)
\(L(\frac12)\) \(\approx\) \(0.549030 + 2.00825i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.468 - 1.33i)T \)
11 \( 1 \)
good3 \( 1 + (-0.127 + 0.393i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (-1.55 - 2.14i)T + (-1.54 + 4.75i)T^{2} \)
7 \( 1 + (-1.01 - 3.12i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-1.10 - 0.801i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (3.22 + 4.44i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.16 - 0.702i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 - 6.38iT - 23T^{2} \)
29 \( 1 + (2.03 + 6.25i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-0.644 + 0.886i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (-4.60 + 1.49i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (9.54 + 3.10i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 - 6.43iT - 43T^{2} \)
47 \( 1 + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-4.39 + 6.05i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.00 - 3.08i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (5.32 - 3.86i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 6.07T + 67T^{2} \)
71 \( 1 + (-3.75 - 5.16i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.895 - 0.291i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (7.52 + 5.47i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (2.67 + 3.67i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + 10.6T + 89T^{2} \)
97 \( 1 + (6.05 + 4.39i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.06043375835499979002665864041, −9.451817193616802518202809020990, −8.579134900816126194042827497066, −7.60402579831793708834126097098, −6.98188034818580229595493203460, −6.07237512723853996052702417139, −5.40154584210946109070757557201, −4.41222160833041869775310407468, −3.01269589983836362839155111145, −2.02607594377501071671567066404, 0.935901505132151969578071689686, 1.76866704490821651382637371663, 3.41644320479774307748960358099, 4.34588218820055001092987294609, 4.87134089549927908463931542257, 6.02256470789207770226432167345, 7.03078242759784812872314537135, 8.421202244920402366460799809331, 8.977156602540985906685611222784, 9.904723093603391400765959769864

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