Properties

Label 2-968-11.10-c0-0-0
Degree $2$
Conductor $968$
Sign $0.426 - 0.904i$
Analytic cond. $0.483094$
Root an. cond. $0.695050$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s + 1.41i·7-s − 15-s + 1.41i·17-s − 1.41i·21-s − 23-s + 27-s + 31-s + 1.41i·35-s + 37-s + 1.41i·43-s − 1.00·49-s − 1.41i·51-s + 59-s + ⋯
L(s)  = 1  − 3-s + 5-s + 1.41i·7-s − 15-s + 1.41i·17-s − 1.41i·21-s − 23-s + 27-s + 31-s + 1.41i·35-s + 37-s + 1.41i·43-s − 1.00·49-s − 1.41i·51-s + 59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.426 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.426 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $0.426 - 0.904i$
Analytic conductor: \(0.483094\)
Root analytic conductor: \(0.695050\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{968} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 968,\ (\ :0),\ 0.426 - 0.904i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7886763449\)
\(L(\frac12)\) \(\approx\) \(0.7886763449\)
\(L(1)\) 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 \( 1 + T + T^{2} \)
5 \( 1 - T + T^{2} \)
7 \( 1 - 1.41iT - T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - 1.41iT - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T + T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 1.41iT - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T + T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + 1.41iT - T^{2} \)
83 \( 1 + 1.41iT - T^{2} \)
89 \( 1 + T + T^{2} \)
97 \( 1 + T + T^{2} \)
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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

−10.29899130671281585511177224199, −9.698767793115177574493064887325, −8.731718541648375726483639070846, −8.043006352600327529359166118573, −6.46429044551159361631351075448, −5.96257813594915694726073057534, −5.55201673953473326287727927844, −4.42751356559661842221418965394, −2.80834934473539215359122660250, −1.76567963821866631400230613918, 0.888355604703299204175892473574, 2.49730313996834084159477406008, 3.97475509500580425100787307385, 4.97351158894988733359554889810, 5.76192932390643374241656158249, 6.59390574852220492798975328233, 7.29596142280420772880598090374, 8.368606414729696413023233107418, 9.647044327939144600882331096177, 10.04854837437942496760166223865

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