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Properties

Label 2-966-7.2-c1-0-27
Degree $2$
Conductor $966$
Sign $-0.991 - 0.126i$
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  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (1.28 − 2.21i)5-s − 0.999·6-s + (0.5 − 2.59i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (−1.28 − 2.21i)10-s + (−1.5 − 2.59i)11-s + (−0.499 + 0.866i)12-s − 3.12·13-s + (−2 − 1.73i)14-s − 2.56·15-s + (−0.5 + 0.866i)16-s + (2.34 + 4.05i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.572 − 0.992i)5-s − 0.408·6-s + (0.188 − 0.981i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.405 − 0.701i)10-s + (−0.452 − 0.783i)11-s + (−0.144 + 0.249i)12-s − 0.866·13-s + (−0.534 − 0.462i)14-s − 0.661·15-s + (−0.125 + 0.216i)16-s + (0.568 + 0.983i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.991 - 0.126i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ -0.991 - 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0979841 + 1.54402i\)
\(L(\frac12)\) \(\approx\) \(0.0979841 + 1.54402i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-0.5 + 2.59i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (-1.28 + 2.21i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.12T + 13T^{2} \)
17 \( 1 + (-2.34 - 4.05i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.56 + 2.70i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 - 0.123T + 29T^{2} \)
31 \( 1 + (-0.842 - 1.45i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.438 - 0.759i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 7.12T + 41T^{2} \)
43 \( 1 - 12.2T + 43T^{2} \)
47 \( 1 + (-0.219 + 0.379i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.280 - 0.486i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.719 + 1.24i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1 - 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.438T + 71T^{2} \)
73 \( 1 + (8.46 + 14.6i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.18 - 10.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.56T + 83T^{2} \)
89 \( 1 + (4.12 - 7.14i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 11.6T + 97T^{2} \)
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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

−9.774267108327263797275078505451, −8.805671074266829006476080658911, −7.964519821773889496251159515870, −7.01251738225130809958189297114, −5.86766842697337339331618211020, −5.17176358291732217979618909008, −4.33658161393679135672321912256, −3.04631096637876966198096573958, −1.67817850869430104443043320724, −0.66463016408914392829976673251, 2.32157030391958868786276447015, 3.12168029932276558979264617427, 4.56344153807107556674003014013, 5.37951881501475106518104942936, 5.98802587236872535905884804597, 7.05344366556708363267752900363, 7.66547058101615812501359869129, 8.867528042180358532798283616980, 9.780249246609300531643717078614, 10.16449483726037681131887614824

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