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Properties

Label 2-930-155.109-c1-0-6
Degree $2$
Conductor $930$
Sign $-0.986 + 0.161i$
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  + (0.587 + 0.809i)2-s + (−0.587 + 0.809i)3-s + (−0.309 + 0.951i)4-s + (1.71 + 1.42i)5-s − 6-s + (−2.04 − 0.664i)7-s + (−0.951 + 0.309i)8-s + (−0.309 − 0.951i)9-s + (−0.145 + 2.23i)10-s + (−0.164 + 0.504i)11-s + (−0.587 − 0.809i)12-s + (−1.53 + 2.11i)13-s + (−0.664 − 2.04i)14-s + (−2.16 + 0.550i)15-s + (−0.809 − 0.587i)16-s + (−2.18 + 0.710i)17-s + ⋯
L(s)  = 1  + (0.415 + 0.572i)2-s + (−0.339 + 0.467i)3-s + (−0.154 + 0.475i)4-s + (0.768 + 0.639i)5-s − 0.408·6-s + (−0.772 − 0.250i)7-s + (−0.336 + 0.109i)8-s + (−0.103 − 0.317i)9-s + (−0.0461 + 0.705i)10-s + (−0.0494 + 0.152i)11-s + (−0.169 − 0.233i)12-s + (−0.426 + 0.587i)13-s + (−0.177 − 0.546i)14-s + (−0.559 + 0.142i)15-s + (−0.202 − 0.146i)16-s + (−0.530 + 0.172i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.161i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 + 0.161i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.986 + 0.161i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.986 + 0.161i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0897254 - 1.10702i\)
\(L(\frac12)\) \(\approx\) \(0.0897254 - 1.10702i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.587 - 0.809i)T \)
3 \( 1 + (0.587 - 0.809i)T \)
5 \( 1 + (-1.71 - 1.42i)T \)
31 \( 1 + (0.566 - 5.53i)T \)
good7 \( 1 + (2.04 + 0.664i)T + (5.66 + 4.11i)T^{2} \)
11 \( 1 + (0.164 - 0.504i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (1.53 - 2.11i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (2.18 - 0.710i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (3.43 - 2.49i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (4.25 - 1.38i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 + (-3.97 + 2.88i)T + (8.96 - 27.5i)T^{2} \)
37 \( 1 + 2.67iT - 37T^{2} \)
41 \( 1 + (-0.305 + 0.221i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + (-0.723 - 0.995i)T + (-13.2 + 40.8i)T^{2} \)
47 \( 1 + (1.80 - 2.47i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (1.05 - 0.343i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (3.02 + 2.19i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 - 1.98T + 61T^{2} \)
67 \( 1 + 2.64iT - 67T^{2} \)
71 \( 1 + (-0.970 - 2.98i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-15.5 - 5.05i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (1.41 + 4.36i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.23 - 3.07i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (0.173 - 0.533i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-4.16 - 1.35i)T + (78.4 + 57.0i)T^{2} \)
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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

−10.34332426436087648930708676770, −9.767359369025679623457384576891, −8.972012635783175620884116970828, −7.82411819767676536884291861297, −6.63615179390933211611731145727, −6.42410326277236523848631936988, −5.40130637439064362150736289397, −4.35677738659711852707670235034, −3.43124906176188890921796194457, −2.18811038809744932556983410287, 0.44077944869824828080473992275, 1.99732564562406123778365962490, 2.88671544661522359886126345008, 4.35441101763904282107575626656, 5.24442349419331758275277683981, 6.10448751670353163551547362396, 6.70315239078151428397906586134, 8.076745901223201428073413442530, 8.965496980250758370935225404955, 9.767305438977717827519738730738

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