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Properties

Label 2-968-88.69-c1-0-28
Degree $2$
Conductor $968$
Sign $-0.430 - 0.902i$
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  + (1.40 + 0.118i)2-s + (−0.305 + 0.0992i)3-s + (1.97 + 0.335i)4-s + (−0.756 + 1.04i)5-s + (−0.442 + 0.103i)6-s + (−1.02 + 3.14i)7-s + (2.73 + 0.707i)8-s + (−2.34 + 1.70i)9-s + (−1.19 + 1.37i)10-s + (−0.635 + 0.0932i)12-s + (−1.81 − 2.49i)13-s + (−1.81 + 4.31i)14-s + (0.127 − 0.393i)15-s + (3.77 + 1.32i)16-s + (2.58 + 1.87i)17-s + (−3.50 + 2.12i)18-s + ⋯
L(s)  = 1  + (0.996 + 0.0841i)2-s + (−0.176 + 0.0573i)3-s + (0.985 + 0.167i)4-s + (−0.338 + 0.465i)5-s + (−0.180 + 0.0422i)6-s + (−0.386 + 1.18i)7-s + (0.968 + 0.250i)8-s + (−0.781 + 0.567i)9-s + (−0.376 + 0.435i)10-s + (−0.183 + 0.0269i)12-s + (−0.503 − 0.692i)13-s + (−0.484 + 1.15i)14-s + (0.0329 − 0.101i)15-s + (0.943 + 0.330i)16-s + (0.627 + 0.455i)17-s + (−0.826 + 0.499i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.430 - 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.430 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $-0.430 - 0.902i$
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: \((2,\ 968,\ (\ :1/2),\ -0.430 - 0.902i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08853 + 1.72534i\)
\(L(\frac12)\) \(\approx\) \(1.08853 + 1.72534i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 - 0.118i)T \)
11 \( 1 \)
good3 \( 1 + (0.305 - 0.0992i)T + (2.42 - 1.76i)T^{2} \)
5 \( 1 + (0.756 - 1.04i)T + (-1.54 - 4.75i)T^{2} \)
7 \( 1 + (1.02 - 3.14i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (1.81 + 2.49i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-2.58 - 1.87i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (4.48 - 1.45i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + 3.47T + 23T^{2} \)
29 \( 1 + (5.88 + 1.91i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-2.16 + 1.57i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-7.46 - 2.42i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.60 - 4.93i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 2.37iT - 43T^{2} \)
47 \( 1 + (-0.986 - 3.03i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (2.77 + 3.82i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-13.4 - 4.37i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-9.14 + 12.5i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 - 4.70iT - 67T^{2} \)
71 \( 1 + (-7.42 - 5.39i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.290 - 0.893i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (7.73 - 5.62i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (0.751 - 1.03i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 - 15.3T + 89T^{2} \)
97 \( 1 + (-11.2 + 8.20i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.49491056782907275550771533403, −9.621685686472209107368041362945, −8.246381889884727878603370070814, −7.83583852152741709573504090196, −6.56656071644864915244295427821, −5.81171763343946220599116041048, −5.28466993098521168274883616857, −4.02014950512170013024915913523, −2.95264395866951286375176678138, −2.23501625839805671353487117265, 0.65780271916621902124069378498, 2.38072162222181256578299885085, 3.68570735013219802890369757446, 4.26198850886087136734830336644, 5.29057218884374523705228001287, 6.29602632338702349146313336843, 7.00373796280594718272451377889, 7.81275028749396878041697180053, 8.941826763471198451764328242799, 9.969546149915145384909219362579

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