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Properties

Label 2-966-161.160-c1-0-6
Degree $2$
Conductor $966$
Sign $-0.0182 - 0.999i$
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  + 2-s − i·3-s + 4-s − 1.36·5-s − i·6-s + (−2.48 + 0.912i)7-s + 8-s − 9-s − 1.36·10-s + 5.58i·11-s − i·12-s + 4.43i·13-s + (−2.48 + 0.912i)14-s + 1.36i·15-s + 16-s − 4.96·17-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577i·3-s + 0.5·4-s − 0.609·5-s − 0.408i·6-s + (−0.938 + 0.344i)7-s + 0.353·8-s − 0.333·9-s − 0.431·10-s + 1.68i·11-s − 0.288i·12-s + 1.22i·13-s + (−0.663 + 0.243i)14-s + 0.351i·15-s + 0.250·16-s − 1.20·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0182 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0182 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.0182 - 0.999i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (643, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ -0.0182 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.915990 + 0.932839i\)
\(L(\frac12)\) \(\approx\) \(0.915990 + 0.932839i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + iT \)
7 \( 1 + (2.48 - 0.912i)T \)
23 \( 1 + (4.53 + 1.57i)T \)
good5 \( 1 + 1.36T + 5T^{2} \)
11 \( 1 - 5.58iT - 11T^{2} \)
13 \( 1 - 4.43iT - 13T^{2} \)
17 \( 1 + 4.96T + 17T^{2} \)
19 \( 1 - 4.68T + 19T^{2} \)
29 \( 1 - 7.94T + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 3.60iT - 37T^{2} \)
41 \( 1 - 8.43iT - 41T^{2} \)
43 \( 1 - 5.42iT - 43T^{2} \)
47 \( 1 + 6.14iT - 47T^{2} \)
53 \( 1 + 5.25iT - 53T^{2} \)
59 \( 1 - 10.3iT - 59T^{2} \)
61 \( 1 - 0.286T + 61T^{2} \)
67 \( 1 + 8.74iT - 67T^{2} \)
71 \( 1 + 14.3T + 71T^{2} \)
73 \( 1 + 6.57iT - 73T^{2} \)
79 \( 1 - 10.8iT - 79T^{2} \)
83 \( 1 + 2.43T + 83T^{2} \)
89 \( 1 + 17.0T + 89T^{2} \)
97 \( 1 - 4.19T + 97T^{2} \)
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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

−10.12748567452552902265632681332, −9.521645305143278281313266289008, −8.446418639586563864132435175815, −7.39744263979721147473681874801, −6.77680723504877908646446352483, −6.16579365704820114923806050437, −4.76284471666693494304091000684, −4.13232459078490112026780664887, −2.84097488477120582202030361885, −1.84554192349600011005050269523, 0.45692993508611996510842782456, 2.84890519563204484286005283255, 3.49217482597933446454769807942, 4.26828033434949593037931076107, 5.56533239549440565053411759909, 6.07161176514672231265549418316, 7.20781750193295379461013792378, 8.127942434314519308708174542294, 8.936090232543509286042003763543, 10.04692608189075075839638738977

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