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Properties

Label 2-966-69.68-c1-0-37
Degree $2$
Conductor $966$
Sign $0.524 + 0.851i$
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 + (1.73 − 0.0164i)3-s − 4-s − 3.75·5-s + (0.0164 + 1.73i)6-s − i·7-s − i·8-s + (2.99 − 0.0568i)9-s − 3.75i·10-s − 2.91·11-s + (−1.73 + 0.0164i)12-s − 1.96·13-s + 14-s + (−6.51 + 0.0616i)15-s + 16-s + 6.79·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.999 − 0.00946i)3-s − 0.5·4-s − 1.68·5-s + (0.00669 + 0.707i)6-s − 0.377i·7-s − 0.353i·8-s + (0.999 − 0.0189i)9-s − 1.18i·10-s − 0.878·11-s + (−0.499 + 0.00473i)12-s − 0.544·13-s + 0.267·14-s + (−1.68 + 0.0159i)15-s + 0.250·16-s + 1.64·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.524 + 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.524 + 0.851i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (827, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ 0.524 + 0.851i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.893581 - 0.499036i\)
\(L(\frac12)\) \(\approx\) \(0.893581 - 0.499036i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 + 0.0164i)T \)
7 \( 1 + iT \)
23 \( 1 + (2.47 + 4.10i)T \)
good5 \( 1 + 3.75T + 5T^{2} \)
11 \( 1 + 2.91T + 11T^{2} \)
13 \( 1 + 1.96T + 13T^{2} \)
17 \( 1 - 6.79T + 17T^{2} \)
19 \( 1 + 5.60iT - 19T^{2} \)
29 \( 1 + 9.69iT - 29T^{2} \)
31 \( 1 + 8.11T + 31T^{2} \)
37 \( 1 + 1.15iT - 37T^{2} \)
41 \( 1 - 1.92iT - 41T^{2} \)
43 \( 1 + 6.91iT - 43T^{2} \)
47 \( 1 + 10.7iT - 47T^{2} \)
53 \( 1 - 2.59T + 53T^{2} \)
59 \( 1 - 6.79iT - 59T^{2} \)
61 \( 1 + 5.64iT - 61T^{2} \)
67 \( 1 - 12.1iT - 67T^{2} \)
71 \( 1 - 4.18iT - 71T^{2} \)
73 \( 1 + 11.1T + 73T^{2} \)
79 \( 1 - 8.47iT - 79T^{2} \)
83 \( 1 + 8.52T + 83T^{2} \)
89 \( 1 + 5.22T + 89T^{2} \)
97 \( 1 - 1.81iT - 97T^{2} \)
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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

−9.787130010565614347608266895756, −8.654598775031370691655278140917, −8.094587743912839119688783498012, −7.41449159249637716725488947768, −7.06377832314655142310117605376, −5.42011885727180057469382052298, −4.37613780495849438296742725462, −3.72938883978783593116150242391, −2.66183236417788921737096345296, −0.43489649862912994954476513213, 1.56414952707183339685686254852, 3.13911323311432629758805102758, 3.45571050165788375813550093704, 4.54014732744315327524418424000, 5.55556706755255492092789263935, 7.41538067938749462487377020747, 7.71801498630947900478825535279, 8.393822549395589233197437380044, 9.340867525132950654210020247366, 10.15970265036475520533467939495

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