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Properties

Label 2-930-15.2-c1-0-35
Degree $2$
Conductor $930$
Sign $-0.0381 + 0.999i$
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  + (−0.707 + 0.707i)2-s + (−1.30 − 1.14i)3-s − 1.00i·4-s + (1.36 − 1.77i)5-s + (1.72 − 0.112i)6-s + (−0.0800 − 0.0800i)7-s + (0.707 + 0.707i)8-s + (0.390 + 2.97i)9-s + (0.286 + 2.21i)10-s − 3.80i·11-s + (−1.14 + 1.30i)12-s + (2.77 − 2.77i)13-s + 0.113·14-s + (−3.80 + 0.745i)15-s − 1.00·16-s + (4.20 − 4.20i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.751 − 0.659i)3-s − 0.500i·4-s + (0.610 − 0.791i)5-s + (0.705 − 0.0461i)6-s + (−0.0302 − 0.0302i)7-s + (0.250 + 0.250i)8-s + (0.130 + 0.991i)9-s + (0.0905 + 0.701i)10-s − 1.14i·11-s + (−0.329 + 0.375i)12-s + (0.768 − 0.768i)13-s + 0.0302·14-s + (−0.981 + 0.192i)15-s − 0.250·16-s + (1.02 − 1.02i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0381 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0381 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.0381 + 0.999i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (497, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.0381 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.689124 - 0.715936i\)
\(L(\frac12)\) \(\approx\) \(0.689124 - 0.715936i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (1.30 + 1.14i)T \)
5 \( 1 + (-1.36 + 1.77i)T \)
31 \( 1 + T \)
good7 \( 1 + (0.0800 + 0.0800i)T + 7iT^{2} \)
11 \( 1 + 3.80iT - 11T^{2} \)
13 \( 1 + (-2.77 + 2.77i)T - 13iT^{2} \)
17 \( 1 + (-4.20 + 4.20i)T - 17iT^{2} \)
19 \( 1 - 6.37iT - 19T^{2} \)
23 \( 1 + (-4.35 - 4.35i)T + 23iT^{2} \)
29 \( 1 + 7.03T + 29T^{2} \)
37 \( 1 + (-3.86 - 3.86i)T + 37iT^{2} \)
41 \( 1 + 9.64iT - 41T^{2} \)
43 \( 1 + (-1.84 + 1.84i)T - 43iT^{2} \)
47 \( 1 + (2.31 - 2.31i)T - 47iT^{2} \)
53 \( 1 + (5.01 + 5.01i)T + 53iT^{2} \)
59 \( 1 + 8.84T + 59T^{2} \)
61 \( 1 + 7.40T + 61T^{2} \)
67 \( 1 + (-3.19 - 3.19i)T + 67iT^{2} \)
71 \( 1 - 0.739iT - 71T^{2} \)
73 \( 1 + (-6.88 + 6.88i)T - 73iT^{2} \)
79 \( 1 + 3.84iT - 79T^{2} \)
83 \( 1 + (4.34 + 4.34i)T + 83iT^{2} \)
89 \( 1 - 15.1T + 89T^{2} \)
97 \( 1 + (4.93 + 4.93i)T + 97iT^{2} \)
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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

−9.795920398903853701389437994251, −8.922262973559258719977340549502, −8.037424802591885924625277319035, −7.47332969139107342675383984095, −6.19294287011062404412519538265, −5.66741059183329269912136712101, −5.13773946987632730196572104554, −3.40982614941618088914408011022, −1.62453340581325132571978853384, −0.67742556034761493506230405126, 1.47306783887883661113922616761, 2.83074051759411052734519772602, 3.96177395493886281792415314222, 4.91097098143799640195106587041, 6.12342764517827631521536643242, 6.75152568545673438471342853134, 7.71312058607543275156925512792, 9.245501758717688306527663631813, 9.381061270672030119192846346165, 10.41597853496042113323188634512

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