Properties

Label 2-966-23.13-c1-0-21
Degree $2$
Conductor $966$
Sign $0.861 - 0.508i$
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  + (0.654 + 0.755i)2-s + (0.841 + 0.540i)3-s + (−0.142 + 0.989i)4-s + (0.722 − 1.58i)5-s + (0.142 + 0.989i)6-s + (0.959 + 0.281i)7-s + (−0.841 + 0.540i)8-s + (0.415 + 0.909i)9-s + (1.66 − 0.490i)10-s + (2.45 − 2.83i)11-s + (−0.654 + 0.755i)12-s + (4.24 − 1.24i)13-s + (0.415 + 0.909i)14-s + (1.46 − 0.940i)15-s + (−0.959 − 0.281i)16-s + (−0.560 − 3.90i)17-s + ⋯
L(s)  = 1  + (0.463 + 0.534i)2-s + (0.485 + 0.312i)3-s + (−0.0711 + 0.494i)4-s + (0.323 − 0.707i)5-s + (0.0580 + 0.404i)6-s + (0.362 + 0.106i)7-s + (−0.297 + 0.191i)8-s + (0.138 + 0.303i)9-s + (0.527 − 0.154i)10-s + (0.739 − 0.853i)11-s + (−0.189 + 0.218i)12-s + (1.17 − 0.345i)13-s + (0.111 + 0.243i)14-s + (0.377 − 0.242i)15-s + (−0.239 − 0.0704i)16-s + (−0.136 − 0.946i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.861 - 0.508i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.861 - 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.861 - 0.508i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (841, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ 0.861 - 0.508i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.66134 + 0.727151i\)
\(L(\frac12)\) \(\approx\) \(2.66134 + 0.727151i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.654 - 0.755i)T \)
3 \( 1 + (-0.841 - 0.540i)T \)
7 \( 1 + (-0.959 - 0.281i)T \)
23 \( 1 + (4.45 - 1.77i)T \)
good5 \( 1 + (-0.722 + 1.58i)T + (-3.27 - 3.77i)T^{2} \)
11 \( 1 + (-2.45 + 2.83i)T + (-1.56 - 10.8i)T^{2} \)
13 \( 1 + (-4.24 + 1.24i)T + (10.9 - 7.02i)T^{2} \)
17 \( 1 + (0.560 + 3.90i)T + (-16.3 + 4.78i)T^{2} \)
19 \( 1 + (-0.461 + 3.20i)T + (-18.2 - 5.35i)T^{2} \)
29 \( 1 + (-0.308 - 2.14i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (4.61 - 2.96i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 + (-1.52 - 3.33i)T + (-24.2 + 27.9i)T^{2} \)
41 \( 1 + (3.11 - 6.81i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (-3.67 - 2.36i)T + (17.8 + 39.1i)T^{2} \)
47 \( 1 - 4.05T + 47T^{2} \)
53 \( 1 + (5.50 + 1.61i)T + (44.5 + 28.6i)T^{2} \)
59 \( 1 + (-4.78 + 1.40i)T + (49.6 - 31.8i)T^{2} \)
61 \( 1 + (-2.95 + 1.89i)T + (25.3 - 55.4i)T^{2} \)
67 \( 1 + (1.13 + 1.31i)T + (-9.53 + 66.3i)T^{2} \)
71 \( 1 + (-0.838 - 0.967i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (1.29 - 9.01i)T + (-70.0 - 20.5i)T^{2} \)
79 \( 1 + (12.1 - 3.56i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (5.69 + 12.4i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (1.19 + 0.766i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (5.34 - 11.6i)T + (-63.5 - 73.3i)T^{2} \)
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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

−9.844199440053442863967463076392, −8.900316312086716645911952650398, −8.636289076438229078705303552824, −7.64720911378441393355150856789, −6.57479673073938696246268363419, −5.65617240935307662456420012218, −4.90280173756925947967881374283, −3.89164115443550845635389161520, −2.96426203253020596338102798928, −1.30790028611732571192029716245, 1.54760575907584764358460118315, 2.30891580527600972877237288714, 3.74288136769664891976095712987, 4.19219814952431539883136147909, 5.78544699864153027017881851335, 6.41973775952798906562450068659, 7.33258330627719446246960225336, 8.372588443419898124397665764768, 9.181992385106295066696789833009, 10.14500315592335335557812300537

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