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Properties

Label 2-966-7.4-c1-0-9
Degree $2$
Conductor $966$
Sign $0.827 - 0.561i$
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 + (0.5 + 0.866i)5-s + 0.999·6-s + (1.62 + 2.09i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (0.499 − 0.866i)10-s + (1.5 − 2.59i)11-s + (−0.499 − 0.866i)12-s + 1.41·13-s + (0.999 − 2.44i)14-s − 0.999·15-s + (−0.5 − 0.866i)16-s + (0.707 − 1.22i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.223 + 0.387i)5-s + 0.408·6-s + (0.612 + 0.790i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.158 − 0.273i)10-s + (0.452 − 0.783i)11-s + (−0.144 − 0.249i)12-s + 0.392·13-s + (0.267 − 0.654i)14-s − 0.258·15-s + (−0.125 − 0.216i)16-s + (0.171 − 0.297i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.827 - 0.561i$
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: \((2,\ 966,\ (\ :1/2),\ 0.827 - 0.561i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27014 + 0.390276i\)
\(L(\frac12)\) \(\approx\) \(1.27014 + 0.390276i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.62 - 2.09i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.41T + 13T^{2} \)
17 \( 1 + (-0.707 + 1.22i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.70 - 2.95i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 + 0.757T + 29T^{2} \)
31 \( 1 + (2.32 - 4.03i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.53 - 7.85i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 1.75T + 41T^{2} \)
43 \( 1 + 7.65T + 43T^{2} \)
47 \( 1 + (-4.65 - 8.06i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.08 + 1.88i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.378 + 0.655i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3 - 5.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.17 + 3.76i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.89T + 71T^{2} \)
73 \( 1 + (2.82 - 4.89i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.207 - 0.358i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 12.3T + 83T^{2} \)
89 \( 1 + (7.36 + 12.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 3.58T + 97T^{2} \)
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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

−10.17427039577949511120288484457, −9.335577784305016876389247207936, −8.624110743011871789759880893759, −7.889212568374968444061252782296, −6.56485399361678175725032078656, −5.72063167563025534286538157232, −4.79306308806139794463628573103, −3.61371532190862804853129675936, −2.70341631340370001113556135879, −1.28847866547259278671972320792, 0.866809536525105088644741031593, 1.95764570977693090815243267452, 3.87961509407812953181201397515, 4.87358620062194285953328035100, 5.67090338692064512379778593471, 6.77807674926413520475378799927, 7.32494003703237601327828990740, 8.128406279334746342612277373612, 9.039239726510541472152817853167, 9.783543971234939263202650407288

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