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Properties

Label 2-968-11.3-c1-0-25
Degree $2$
Conductor $968$
Sign $-0.751 - 0.659i$
Analytic cond. $7.72951$
Root an. cond. $2.78020$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 1.53i)3-s + (−0.5 − 0.363i)5-s + (−0.5 + 1.53i)7-s + (0.309 − 0.224i)9-s + (−4.73 + 3.44i)13-s + (−0.309 + 0.951i)15-s + (−1.5 − 1.08i)17-s + (−1.5 − 4.61i)19-s + 2.61·21-s − 4·23-s + (−1.42 − 4.39i)25-s + (−4.42 − 3.21i)27-s + (−2.26 + 6.96i)29-s + (−0.881 + 0.640i)31-s + (0.809 − 0.587i)35-s + ⋯
L(s)  = 1  + (−0.288 − 0.888i)3-s + (−0.223 − 0.162i)5-s + (−0.188 + 0.581i)7-s + (0.103 − 0.0748i)9-s + (−1.31 + 0.954i)13-s + (−0.0797 + 0.245i)15-s + (−0.363 − 0.264i)17-s + (−0.344 − 1.05i)19-s + 0.571·21-s − 0.834·23-s + (−0.285 − 0.878i)25-s + (−0.851 − 0.619i)27-s + (−0.420 + 1.29i)29-s + (−0.158 + 0.115i)31-s + (0.136 − 0.0993i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.751 - 0.659i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.751 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $-0.751 - 0.659i$
Analytic conductor: \(7.72951\)
Root analytic conductor: \(2.78020\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{968} (729, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 968,\ (\ :1/2),\ -0.751 - 0.659i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
good3 \( 1 + (0.5 + 1.53i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (0.5 + 0.363i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.5 - 1.53i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (4.73 - 3.44i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.5 + 1.08i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.5 + 4.61i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + (2.26 - 6.96i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.881 - 0.640i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.97 - 9.14i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.97 - 9.14i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 1.52T + 43T^{2} \)
47 \( 1 + (3.26 + 10.0i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.5 - 0.363i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.736 - 2.26i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (1.5 + 1.08i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 14.4T + 67T^{2} \)
71 \( 1 + (-4.11 - 2.99i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.972 + 2.99i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (3.11 - 2.26i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-7.59 - 5.51i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 4.47T + 89T^{2} \)
97 \( 1 + (-9.97 + 7.24i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.436471095503702999469326868188, −8.682853030771989826920917009880, −7.66607476509033615171833985476, −6.86442228364330377677744071102, −6.35484226158552969547950082870, −5.10456946854903446936287782489, −4.28043760848348117393122192835, −2.74198580180955135330202281828, −1.72904588719877146061539163238, 0, 2.12700484838215989704490420027, 3.62351004592595270252987834504, 4.22700366769249206308877111154, 5.27476836842010460240682721543, 6.06458595302149503602470180379, 7.48979605308290102899774872468, 7.70630925125443753922842939066, 9.115879883425577742263852894553, 9.930242040916593437310535614005

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