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Properties

Label 2-968-88.69-c1-0-96
Degree $2$
Conductor $968$
Sign $-0.805 + 0.592i$
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  + (0.323 − 1.37i)2-s + (2.66 − 0.864i)3-s + (−1.79 − 0.889i)4-s + (0.681 − 0.937i)5-s + (−0.330 − 3.94i)6-s + (0.935 − 2.88i)7-s + (−1.80 + 2.17i)8-s + (3.90 − 2.83i)9-s + (−1.07 − 1.24i)10-s + (−5.53 − 0.819i)12-s + (2.43 + 3.35i)13-s + (−3.66 − 2.21i)14-s + (1.00 − 3.08i)15-s + (2.41 + 3.18i)16-s + (−5.91 − 4.29i)17-s + (−2.64 − 6.29i)18-s + ⋯
L(s)  = 1  + (0.228 − 0.973i)2-s + (1.53 − 0.499i)3-s + (−0.895 − 0.444i)4-s + (0.304 − 0.419i)5-s + (−0.134 − 1.60i)6-s + (0.353 − 1.08i)7-s + (−0.637 + 0.770i)8-s + (1.30 − 0.946i)9-s + (−0.338 − 0.392i)10-s + (−1.59 − 0.236i)12-s + (0.675 + 0.929i)13-s + (−0.978 − 0.593i)14-s + (0.258 − 0.796i)15-s + (0.604 + 0.796i)16-s + (−1.43 − 1.04i)17-s + (−0.623 − 1.48i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.805 + 0.592i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $-0.805 + 0.592i$
Analytic conductor: \(7.72951\)
Root analytic conductor: \(2.78020\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{968} (245, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 968,\ (\ :1/2),\ -0.805 + 0.592i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.898305 - 2.73604i\)
\(L(\frac12)\) \(\approx\) \(0.898305 - 2.73604i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.323 + 1.37i)T \)
11 \( 1 \)
good3 \( 1 + (-2.66 + 0.864i)T + (2.42 - 1.76i)T^{2} \)
5 \( 1 + (-0.681 + 0.937i)T + (-1.54 - 4.75i)T^{2} \)
7 \( 1 + (-0.935 + 2.88i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-2.43 - 3.35i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (5.91 + 4.29i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (4.54 - 1.47i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 - 5.24T + 23T^{2} \)
29 \( 1 + (-2.91 - 0.946i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-5.18 + 3.77i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (6.42 + 2.08i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.387 - 1.19i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 7.39iT - 43T^{2} \)
47 \( 1 + (-0.255 - 0.787i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-4.65 - 6.40i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (4.86 + 1.58i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (3.07 - 4.22i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + 2.79iT - 67T^{2} \)
71 \( 1 + (6.13 + 4.45i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.387 + 1.19i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-11.8 + 8.59i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-0.527 + 0.725i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 - 0.656T + 89T^{2} \)
97 \( 1 + (9.01 - 6.54i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.398478436045019222288954120175, −9.016378525452755364690814166044, −8.348156402875822354886399337393, −7.31127864429464717992881153679, −6.42479006792734979554711726290, −4.72018671176775847456155925500, −4.17550768458760360405175918933, −3.08823079701569736648823407845, −2.07028410440060129727936869544, −1.15567678870239148152238745885, 2.25452191595476906058127970767, 3.10948963548117717998910694160, 4.14583749293711229762925854206, 5.07110003313797672032027375147, 6.19606524425413130386852872459, 6.93905466789872705929829718435, 8.323701890671141353876637317097, 8.517140455488490884813767534716, 8.982566865898762061971764837731, 10.14238282750302611733480449032

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