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Properties

Label 2-966-21.20-c1-0-27
Degree $2$
Conductor $966$
Sign $0.863 + 0.504i$
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 + (−0.740 + 1.56i)3-s − 4-s − 3.67·5-s + (−1.56 − 0.740i)6-s + (−0.229 + 2.63i)7-s − i·8-s + (−1.90 − 2.31i)9-s − 3.67i·10-s + 3.25i·11-s + (0.740 − 1.56i)12-s − 0.990i·13-s + (−2.63 − 0.229i)14-s + (2.72 − 5.75i)15-s + 16-s − 6.53·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.427 + 0.903i)3-s − 0.5·4-s − 1.64·5-s + (−0.639 − 0.302i)6-s + (−0.0866 + 0.996i)7-s − 0.353i·8-s + (−0.634 − 0.773i)9-s − 1.16i·10-s + 0.981i·11-s + (0.213 − 0.451i)12-s − 0.274i·13-s + (−0.704 − 0.0613i)14-s + (0.703 − 1.48i)15-s + 0.250·16-s − 1.58·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.863 + 0.504i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.863 + 0.504i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.863 + 0.504i$
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: \((2,\ 966,\ (\ :1/2),\ 0.863 + 0.504i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0656336 - 0.0177696i\)
\(L(\frac12)\) \(\approx\) \(0.0656336 - 0.0177696i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.740 - 1.56i)T \)
7 \( 1 + (0.229 - 2.63i)T \)
23 \( 1 + iT \)
good5 \( 1 + 3.67T + 5T^{2} \)
11 \( 1 - 3.25iT - 11T^{2} \)
13 \( 1 + 0.990iT - 13T^{2} \)
17 \( 1 + 6.53T + 17T^{2} \)
19 \( 1 + 1.80iT - 19T^{2} \)
29 \( 1 + 1.83iT - 29T^{2} \)
31 \( 1 - 7.84iT - 31T^{2} \)
37 \( 1 + 0.190T + 37T^{2} \)
41 \( 1 - 2.12T + 41T^{2} \)
43 \( 1 - 6.53T + 43T^{2} \)
47 \( 1 - 11.1T + 47T^{2} \)
53 \( 1 + 7.60iT - 53T^{2} \)
59 \( 1 - 4.25T + 59T^{2} \)
61 \( 1 + 3.34iT - 61T^{2} \)
67 \( 1 + 3.26T + 67T^{2} \)
71 \( 1 - 2.80iT - 71T^{2} \)
73 \( 1 + 0.439iT - 73T^{2} \)
79 \( 1 + 5.24T + 79T^{2} \)
83 \( 1 + 16.5T + 83T^{2} \)
89 \( 1 + 8.22T + 89T^{2} \)
97 \( 1 + 12.5iT - 97T^{2} \)
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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

−9.822104634604506493425723197368, −8.832629724172994096026079104644, −8.514258442041482980673686602161, −7.32042052239667016822107553455, −6.62893083956475215251748687734, −5.47881252336714606610421741064, −4.58485201761011176188574042376, −4.07194358025730547024055046488, −2.79731015618258180487323616177, −0.04319418307139159918507410853, 0.977790950785118255842100539517, 2.63748244794303003648241692706, 3.88727956376097012089740980526, 4.41227655405677929452230008613, 5.83246546382807884663239001475, 6.92482057541316920660860077844, 7.59492702368274012592130031571, 8.300286629191910941885402887733, 9.110747179026495345354971060569, 10.62738337817603926957656568330

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