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Properties

Label 2-966-7.4-c1-0-7
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: $\chi_{966} (277, \cdot )$
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

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

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