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Properties

Label 2-966-21.17-c1-0-13
Degree $2$
Conductor $966$
Sign $0.553 - 0.832i$
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  + (−0.866 + 0.5i)2-s + (1.5 + 0.866i)3-s + (0.499 − 0.866i)4-s + (−1.73 − 3i)5-s − 1.73·6-s + (−2.5 + 0.866i)7-s + 0.999i·8-s + (1.5 + 2.59i)9-s + (3 + 1.73i)10-s + (2.59 + 1.5i)11-s + (1.49 − 0.866i)12-s + 3.46i·13-s + (1.73 − 2i)14-s − 6i·15-s + (−0.5 − 0.866i)16-s + (−0.866 + 1.5i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.866 + 0.499i)3-s + (0.249 − 0.433i)4-s + (−0.774 − 1.34i)5-s − 0.707·6-s + (−0.944 + 0.327i)7-s + 0.353i·8-s + (0.5 + 0.866i)9-s + (0.948 + 0.547i)10-s + (0.783 + 0.452i)11-s + (0.433 − 0.250i)12-s + 0.960i·13-s + (0.462 − 0.534i)14-s − 1.54i·15-s + (−0.125 − 0.216i)16-s + (−0.210 + 0.363i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.553 - 0.832i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (185, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ 0.553 - 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13510 + 0.608240i\)
\(L(\frac12)\) \(\approx\) \(1.13510 + 0.608240i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (-1.5 - 0.866i)T \)
7 \( 1 + (2.5 - 0.866i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
good5 \( 1 + (1.73 + 3i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.59 - 1.5i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 + (0.866 - 1.5i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6 + 3.46i)T + (9.5 - 16.4i)T^{2} \)
29 \( 1 + 3iT - 29T^{2} \)
31 \( 1 + (-3 - 1.73i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.46T + 41T^{2} \)
43 \( 1 - 10T + 43T^{2} \)
47 \( 1 + (-4.33 - 7.5i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-10.3 - 6i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.92 - 12i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (12 - 6.92i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9iT - 71T^{2} \)
73 \( 1 + (-7.5 - 4.33i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.5 + 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.46T + 83T^{2} \)
89 \( 1 + (-3.46 - 6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 3.46iT - 97T^{2} \)
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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

−9.594089149717603762554686568381, −9.145120568314567971912566291554, −8.854616768823750151792362697803, −7.74932224643318475895741532008, −7.10113853968234431828198042677, −5.89447425439026169374913450361, −4.63125956071794473475543601928, −4.07750160909396446809897453633, −2.72758457602605648732733635639, −1.17874714926869710279135177254, 0.807660225016461357025373535174, 2.60164062846492449596037687328, 3.35813316044301760055809008869, 3.80610578686650108355649272186, 5.98566536918012209530376388698, 6.90604776026748940453043342288, 7.41074885690787754264736768583, 8.095592292328104894773239374604, 9.147670743555484905863014052787, 9.839579367788683344449430228858

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