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Properties

Label 2-966-7.4-c1-0-12
Degree $2$
Conductor $966$
Sign $0.308 + 0.951i$
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 + (−0.914 − 1.58i)5-s + 0.999·6-s + (2.45 − 0.974i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (−0.914 + 1.58i)10-s + (−0.840 + 1.45i)11-s + (−0.499 − 0.866i)12-s + 3.77·13-s + (−2.07 − 1.64i)14-s + 1.82·15-s + (−0.5 − 0.866i)16-s + (−1.29 + 2.25i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.408 − 0.708i)5-s + 0.408·6-s + (0.929 − 0.368i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.289 + 0.500i)10-s + (−0.253 + 0.438i)11-s + (−0.144 − 0.249i)12-s + 1.04·13-s + (−0.554 − 0.439i)14-s + 0.472·15-s + (−0.125 − 0.216i)16-s + (−0.315 + 0.546i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.308 + 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.308 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.308 + 0.951i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ 0.308 + 0.951i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.923423 - 0.671069i\)
\(L(\frac12)\) \(\approx\) \(0.923423 - 0.671069i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.45 + 0.974i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (0.914 + 1.58i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.840 - 1.45i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.77T + 13T^{2} \)
17 \( 1 + (1.29 - 2.25i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.545 + 0.944i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 + 0.771T + 29T^{2} \)
31 \( 1 + (-1.93 + 3.34i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.22 + 2.12i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 1.09T + 41T^{2} \)
43 \( 1 - 10.2T + 43T^{2} \)
47 \( 1 + (-1.11 - 1.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.68 + 6.38i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.21 + 5.56i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.54 + 7.86i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.44 + 11.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.22T + 71T^{2} \)
73 \( 1 + (-1.43 + 2.47i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.385 + 0.668i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.33T + 83T^{2} \)
89 \( 1 + (1.60 + 2.77i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 10.4T + 97T^{2} \)
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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

−9.952019720615709268480187360051, −9.000145627367627270844322292370, −8.348785939674872664316692185890, −7.68719651850560749413811759167, −6.40163951847097167023934730545, −5.17736978871949657473289722536, −4.39619964438396271567063641723, −3.72488819047073102709815099706, −2.08601760294220875165138621527, −0.75830558979744123918235720808, 1.19067730974084589867281068616, 2.65228592326247614451451138429, 4.03931753715583442161548906214, 5.26336028131067417219631412965, 5.98283143960693424887050899552, 6.92847401419178877434916028932, 7.60962792358454636813348484053, 8.411534058246626409075473938941, 9.013185210232440813677208837723, 10.35221245440133950485462779853

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