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Properties

Label 2-930-93.68-c1-0-20
Degree $2$
Conductor $930$
Sign $-0.798 - 0.602i$
Analytic cond. $7.42608$
Root an. cond. $2.72508$
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  + i·2-s + (0.866 + 1.5i)3-s − 4-s + (0.866 + 0.5i)5-s + (−1.5 + 0.866i)6-s + (0.5 + 0.866i)7-s − i·8-s + (−1.5 + 2.59i)9-s + (−0.5 + 0.866i)10-s + (0.866 − 1.5i)11-s + (−0.866 − 1.5i)12-s + (6 + 3.46i)13-s + (−0.866 + 0.5i)14-s + 1.73i·15-s + 16-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.499 + 0.866i)3-s − 0.5·4-s + (0.387 + 0.223i)5-s + (−0.612 + 0.353i)6-s + (0.188 + 0.327i)7-s − 0.353i·8-s + (−0.5 + 0.866i)9-s + (−0.158 + 0.273i)10-s + (0.261 − 0.452i)11-s + (−0.249 − 0.433i)12-s + (1.66 + 0.960i)13-s + (−0.231 + 0.133i)14-s + 0.447i·15-s + 0.250·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.798 - 0.602i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.798 - 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.798 - 0.602i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.798 - 0.602i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.620551 + 1.85370i\)
\(L(\frac12)\) \(\approx\) \(0.620551 + 1.85370i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (-0.866 - 1.5i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
31 \( 1 + (5.5 - 0.866i)T \)
good7 \( 1 + (-0.5 - 0.866i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.866 + 1.5i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-6 - 3.46i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 5.19T + 29T^{2} \)
37 \( 1 + (3 - 1.73i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3 + 1.73i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 6iT - 47T^{2} \)
53 \( 1 + (-0.866 + 1.5i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.59 - 1.5i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 3.46iT - 61T^{2} \)
67 \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-6 - 3.46i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-9 + 5.19i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.06 + 10.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 13.8T + 89T^{2} \)
97 \( 1 + T + 97T^{2} \)
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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

−10.28713619856137285217404575419, −9.204724708558072204728073716928, −8.883851088343908879331625429252, −8.082456365044019681398376858150, −6.96205072517934899884082818252, −5.96047659512279719955890125702, −5.33624226306567039393140566528, −4.06881064201459951498521000381, −3.42743978609614313930242761098, −1.83271575125807837432533801377, 0.949771475103550467207921256036, 1.90433603796962316475727073809, 3.16183150049974410422541626558, 4.02188161281827278587317527763, 5.43406098291162095928695456054, 6.25265413389042287008658038703, 7.37672930685797653753940606887, 8.143226258333325712173679094222, 8.988478196125720127643823071906, 9.575720411928472473615402152217

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