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Properties

Label 2-930-31.5-c1-0-7
Degree $2$
Conductor $930$
Sign $0.695 - 0.718i$
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  + 2-s + (−0.5 + 0.866i)3-s + 4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + 8-s + (−0.499 − 0.866i)9-s + (−0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + (−0.5 + 0.866i)12-s + (3 + 5.19i)13-s + 0.999·15-s + 16-s + (0.5 − 0.866i)17-s + (−0.499 − 0.866i)18-s + (−2 + 3.46i)19-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.288 + 0.499i)3-s + 0.5·4-s + (−0.223 − 0.387i)5-s + (−0.204 + 0.353i)6-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.158 − 0.273i)10-s + (0.150 + 0.261i)11-s + (−0.144 + 0.249i)12-s + (0.832 + 1.44i)13-s + 0.258·15-s + 0.250·16-s + (0.121 − 0.210i)17-s + (−0.117 − 0.204i)18-s + (−0.458 + 0.794i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.695 - 0.718i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.695 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.695 - 0.718i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (811, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.695 - 0.718i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.03893 + 0.864751i\)
\(L(\frac12)\) \(\approx\) \(2.03893 + 0.864751i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (-2 + 5.19i)T \)
good7 \( 1 + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3 - 5.19i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2 - 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 7T + 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
37 \( 1 + (3.5 - 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1 - 1.73i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.5 + 6.06i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3T + 47T^{2} \)
53 \( 1 + (-2 - 3.46i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 14T + 61T^{2} \)
67 \( 1 + (2.5 + 4.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1 + 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.5 - 2.59i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (8 + 13.8i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 4T + 89T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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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

−10.38358508955195293006385249304, −9.265983887851629742557743782328, −8.681136048549815276933184667397, −7.48284618114923384848553218494, −6.52919672020546172684816452615, −5.80717677167120521784200319955, −4.59919520368904898218184099730, −4.21290607583222055117696661273, −3.00785757185597485027405220046, −1.44378102724780495009563903983, 0.998065024629973811541892675766, 2.68314329674674070251577371413, 3.46724156998708297664841128708, 4.75764642334540718625258009648, 5.65169968243665552542141871900, 6.47854961586060048717879920478, 7.20575726398531714710647108545, 8.151449732357854426048694566240, 8.938021436634257134779824726231, 10.45078034470625065355973118830

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