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Properties

Label 2-966-21.20-c1-0-49
Degree $2$
Conductor $966$
Sign $0.526 + 0.850i$
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  + i·2-s + (0.436 − 1.67i)3-s − 4-s + 3.48·5-s + (1.67 + 0.436i)6-s + (−1.82 + 1.91i)7-s − i·8-s + (−2.61 − 1.46i)9-s + 3.48i·10-s − 4.19i·11-s + (−0.436 + 1.67i)12-s − 5.14i·13-s + (−1.91 − 1.82i)14-s + (1.52 − 5.84i)15-s + 16-s − 3.35·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.252 − 0.967i)3-s − 0.5·4-s + 1.55·5-s + (0.684 + 0.178i)6-s + (−0.689 + 0.723i)7-s − 0.353i·8-s + (−0.872 − 0.488i)9-s + 1.10i·10-s − 1.26i·11-s + (−0.126 + 0.483i)12-s − 1.42i·13-s + (−0.511 − 0.487i)14-s + (0.392 − 1.50i)15-s + 0.250·16-s − 0.813·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.526 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.526 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.526 + 0.850i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ 0.526 + 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.51857 - 0.845654i\)
\(L(\frac12)\) \(\approx\) \(1.51857 - 0.845654i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (-0.436 + 1.67i)T \)
7 \( 1 + (1.82 - 1.91i)T \)
23 \( 1 - iT \)
good5 \( 1 - 3.48T + 5T^{2} \)
11 \( 1 + 4.19iT - 11T^{2} \)
13 \( 1 + 5.14iT - 13T^{2} \)
17 \( 1 + 3.35T + 17T^{2} \)
19 \( 1 + 6.99iT - 19T^{2} \)
29 \( 1 + 1.32iT - 29T^{2} \)
31 \( 1 - 8.35iT - 31T^{2} \)
37 \( 1 - 10.1T + 37T^{2} \)
41 \( 1 - 6.46T + 41T^{2} \)
43 \( 1 + 0.124T + 43T^{2} \)
47 \( 1 - 0.718T + 47T^{2} \)
53 \( 1 + 9.13iT - 53T^{2} \)
59 \( 1 + 1.13T + 59T^{2} \)
61 \( 1 - 11.7iT - 61T^{2} \)
67 \( 1 - 0.852T + 67T^{2} \)
71 \( 1 + 8.48iT - 71T^{2} \)
73 \( 1 - 4.79iT - 73T^{2} \)
79 \( 1 - 5.87T + 79T^{2} \)
83 \( 1 + 17.0T + 83T^{2} \)
89 \( 1 - 5.57T + 89T^{2} \)
97 \( 1 + 8.82iT - 97T^{2} \)
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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

−9.524101848397821866690913174724, −8.915748946598886753604832461340, −8.349597003635926466386744253202, −7.15846813083652655900564817581, −6.28366006400345093030678309912, −5.90040832335468036238680500536, −5.14950039723961461426626912807, −3.10038692731908179776777954375, −2.47340057793661931718699799658, −0.77339481492822651085448662724, 1.79652260239379235676609308188, 2.58274273764383894164346508398, 4.05830266908899091264200972618, 4.48486003678395539822334107375, 5.78510767030697567160791120553, 6.51686443511915203281174896825, 7.77270049169342931000568097051, 9.172896621067934584995259482887, 9.503652904678305893876399018312, 9.992083374131721477792352948832

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