Properties

Label 2-966-7.2-c1-0-22
Degree $2$
Conductor $966$
Sign $-0.266 + 0.963i$
Analytic cond. $7.71354$
Root an. cond. $2.77732$
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  + (−0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (1.5 − 2.59i)5-s − 0.999·6-s + (−2.5 − 0.866i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (1.5 + 2.59i)10-s + (−0.5 − 0.866i)11-s + (0.499 − 0.866i)12-s − 6·13-s + (2 − 1.73i)14-s + 3·15-s + (−0.5 + 0.866i)16-s + (1 + 1.73i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.670 − 1.16i)5-s − 0.408·6-s + (−0.944 − 0.327i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (0.474 + 0.821i)10-s + (−0.150 − 0.261i)11-s + (0.144 − 0.249i)12-s − 1.66·13-s + (0.534 − 0.462i)14-s + 0.774·15-s + (−0.125 + 0.216i)16-s + (0.242 + 0.420i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.266 + 0.963i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ -0.266 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.343477 - 0.451495i\)
\(L(\frac12)\) \(\approx\) \(0.343477 - 0.451495i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (2.5 + 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 6T + 13T^{2} \)
17 \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1 + 1.73i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 + 7T + 29T^{2} \)
31 \( 1 + (3.5 + 6.06i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1 + 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.5 + 9.52i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.5 + 12.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-5 - 8.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.5 + 9.52i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 13T + 83T^{2} \)
89 \( 1 + (4 - 6.92i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - T + 97T^{2} \)
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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

−9.487620842230493111988887035186, −9.270344512506887448638965920088, −8.150304588394075340463508576525, −7.37049190507704357889919229485, −6.30777994158210965966384176577, −5.34602186115786228687311210255, −4.76075201385361770602888148257, −3.50795430069871278115690057824, −2.04828039071573666403744488370, −0.26067682897197828543551700295, 1.92237527958281162874417283040, 2.76383080759142530179728279853, 3.42561085167064956658946670582, 5.09353705955192793856188714465, 6.15555771762743253557862360701, 7.14320863162329243498013726139, 7.48257196277459954240382371188, 8.870372306342280414195622581316, 9.676213546611871581085748678867, 10.05589631496281501368192555717

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