Properties

Label 2-930-93.26-c1-0-32
Degree $2$
Conductor $930$
Sign $-0.156 + 0.987i$
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 + (1.68 − 0.395i)3-s − 4-s + (−0.866 + 0.5i)5-s + (−0.395 − 1.68i)6-s + (−1.28 + 2.22i)7-s + i·8-s + (2.68 − 1.33i)9-s + (0.5 + 0.866i)10-s + (−2.56 − 4.44i)11-s + (−1.68 + 0.395i)12-s + (4.59 − 2.65i)13-s + (2.22 + 1.28i)14-s + (−1.26 + 1.18i)15-s + 16-s + (2.59 − 4.49i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.973 − 0.228i)3-s − 0.5·4-s + (−0.387 + 0.223i)5-s + (−0.161 − 0.688i)6-s + (−0.486 + 0.842i)7-s + 0.353i·8-s + (0.895 − 0.444i)9-s + (0.158 + 0.273i)10-s + (−0.774 − 1.34i)11-s + (−0.486 + 0.114i)12-s + (1.27 − 0.735i)13-s + (0.595 + 0.343i)14-s + (−0.325 + 0.306i)15-s + 0.250·16-s + (0.629 − 1.08i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.156 + 0.987i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.156 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19587 - 1.39980i\)
\(L(\frac12)\) \(\approx\) \(1.19587 - 1.39980i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.68 + 0.395i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
31 \( 1 + (-0.790 + 5.51i)T \)
good7 \( 1 + (1.28 - 2.22i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.56 + 4.44i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.59 + 2.65i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.59 + 4.49i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.264 - 0.458i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 3.03T + 23T^{2} \)
29 \( 1 - 8.44T + 29T^{2} \)
37 \( 1 + (-0.580 - 0.335i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-9.85 + 5.68i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.69 + 3.86i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 6.75iT - 47T^{2} \)
53 \( 1 + (0.822 + 1.42i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (10.4 + 6.03i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 7.06iT - 61T^{2} \)
67 \( 1 + (-3.98 - 6.89i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (8.45 - 4.88i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.21 - 1.85i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8.75 - 5.05i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.97 - 6.87i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 14.8T + 89T^{2} \)
97 \( 1 + 6.46T + 97T^{2} \)
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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

−9.828837455068276875131314930721, −8.923943490506628206919698072619, −8.302597848538350907639012803328, −7.71285932397124012639914558711, −6.29878967979187111432593436692, −5.50878080160960629109949259463, −4.03077903583713429414610631567, −3.05173258605066781329148614541, −2.67553142441107073190025572330, −0.854468408849696172655494559852, 1.53984415476898107354531940710, 3.21443834991166630094257636450, 4.15182241593301157505986403664, 4.72713738255230226576157270416, 6.25114761020524842184946701126, 7.03953881922641736949326007191, 7.893381250855685689836307221613, 8.393750063952909490697233031916, 9.366956518617655039021636901095, 10.16402330591034365904168868004

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