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Properties

Label 2-966-7.2-c1-0-13
Degree $2$
Conductor $966$
Sign $0.789 + 0.613i$
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 + (−0.914 + 1.58i)5-s + 0.999·6-s + (−1.75 + 1.98i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (−0.914 − 1.58i)10-s + (−2.07 − 3.59i)11-s + (−0.499 + 0.866i)12-s − 2.18·13-s + (−0.840 − 2.50i)14-s + 1.82·15-s + (−0.5 + 0.866i)16-s + (1.67 + 2.90i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.408 + 0.708i)5-s + 0.408·6-s + (−0.662 + 0.749i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.289 − 0.500i)10-s + (−0.625 − 1.08i)11-s + (−0.144 + 0.249i)12-s − 0.606·13-s + (−0.224 − 0.670i)14-s + 0.472·15-s + (−0.125 + 0.216i)16-s + (0.407 + 0.705i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.789 + 0.613i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.789 + 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.789 + 0.613i$
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: \((2,\ 966,\ (\ :1/2),\ 0.789 + 0.613i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.621848 - 0.213222i\)
\(L(\frac12)\) \(\approx\) \(0.621848 - 0.213222i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.75 - 1.98i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (0.914 - 1.58i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.07 + 3.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.18T + 13T^{2} \)
17 \( 1 + (-1.67 - 2.90i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.66 + 6.35i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 - 5.18T + 29T^{2} \)
31 \( 1 + (5.25 + 9.10i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.518 + 0.898i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 7.33T + 41T^{2} \)
43 \( 1 - 5.36T + 43T^{2} \)
47 \( 1 + (3.61 - 6.25i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.27 + 3.93i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.235 - 0.408i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.37 + 12.7i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.70 + 11.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 14.1T + 71T^{2} \)
73 \( 1 + (5.75 + 9.97i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.59 + 4.49i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.80T + 83T^{2} \)
89 \( 1 + (3.34 - 5.79i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 4.47T + 97T^{2} \)
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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

−9.735162637852598745500186981574, −9.084556091431076798879289739174, −7.999953909115575382471497824707, −7.47328704085766744686598120769, −6.51100819280206506680418835612, −5.87243438007430880802868018985, −5.01434136313558419355978982725, −3.39354411680838417828043531353, −2.47484280175996843514065575478, −0.43884181232487112967781679404, 1.04541315029409704312664101336, 2.75302580880794212059682203865, 3.86145370227337777610812007763, 4.65937491707366140572578880721, 5.49386263149513571128867891296, 6.99612262903042060864038068651, 7.62871834992773041149032985148, 8.593162020467933326599645627667, 9.626584428031052179437903653387, 10.04930023271279652958242693518

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