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Properties

Label 2-968-88.5-c1-0-43
Degree $2$
Conductor $968$
Sign $-0.895 - 0.445i$
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  + (0.636 + 1.26i)2-s + (−0.510 + 0.702i)3-s + (−1.18 + 1.60i)4-s + (3.49 + 1.13i)5-s + (−1.21 − 0.197i)6-s + (0.967 − 0.702i)7-s + (−2.78 − 0.477i)8-s + (0.693 + 2.13i)9-s + (0.791 + 5.13i)10-s + (−0.523 − 1.65i)12-s + (−3.78 + 1.23i)13-s + (1.50 + 0.773i)14-s + (−2.58 + 1.87i)15-s + (−1.17 − 3.82i)16-s + (−0.982 + 3.02i)17-s + (−2.25 + 2.23i)18-s + ⋯
L(s)  = 1  + (0.450 + 0.892i)2-s + (−0.294 + 0.405i)3-s + (−0.594 + 0.804i)4-s + (1.56 + 0.507i)5-s + (−0.495 − 0.0805i)6-s + (0.365 − 0.265i)7-s + (−0.985 − 0.168i)8-s + (0.231 + 0.711i)9-s + (0.250 + 1.62i)10-s + (−0.151 − 0.478i)12-s + (−1.04 + 0.341i)13-s + (0.401 + 0.206i)14-s + (−0.666 + 0.484i)15-s + (−0.293 − 0.956i)16-s + (−0.238 + 0.733i)17-s + (−0.531 + 0.527i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 - 0.445i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.895 - 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $-0.895 - 0.445i$
Analytic conductor: \(7.72951\)
Root analytic conductor: \(2.78020\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{968} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 968,\ (\ :1/2),\ -0.895 - 0.445i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.489956 + 2.08451i\)
\(L(\frac12)\) \(\approx\) \(0.489956 + 2.08451i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.636 - 1.26i)T \)
11 \( 1 \)
good3 \( 1 + (0.510 - 0.702i)T + (-0.927 - 2.85i)T^{2} \)
5 \( 1 + (-3.49 - 1.13i)T + (4.04 + 2.93i)T^{2} \)
7 \( 1 + (-0.967 + 0.702i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (3.78 - 1.23i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.982 - 3.02i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-2.53 + 3.48i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 - 2.46T + 23T^{2} \)
29 \( 1 + (-2.25 - 3.10i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.82 - 8.70i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (3.13 + 4.31i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (0.431 + 0.313i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 6.51iT - 43T^{2} \)
47 \( 1 + (0.290 + 0.210i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (2.30 - 0.749i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-0.350 - 0.481i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (6.93 + 2.25i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 + 11.2iT - 67T^{2} \)
71 \( 1 + (-1.72 + 5.31i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-2.16 + 1.57i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.55 + 10.9i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-0.953 - 0.309i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 - 13.6T + 89T^{2} \)
97 \( 1 + (1.83 + 5.65i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.35873725409322176211239495766, −9.479907138351007113512848627696, −8.791270545879131129660663041937, −7.53879934947249018886321929098, −6.88942032146918323022625647758, −6.05333708221507243201010891758, −5.04223681447405962281708618632, −4.76000474383747516935199056867, −3.16451468940214955965842090654, −1.97730085588646375457205993661, 0.920619323229168781959845714268, 1.97435244721493742993010532392, 2.91490397953545108592514527053, 4.46620915871621069375362595287, 5.31881001488564647476459602637, 5.90943981034674353258623044523, 6.80477563743140633522555384944, 8.168495300600332733605950170805, 9.374761612193595275603477210416, 9.641924200886668119968473416177

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