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Properties

Label 2-930-15.2-c1-0-57
Degree $2$
Conductor $930$
Sign $-0.927 + 0.374i$
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 + (1.37 − 1.05i)3-s − 1.00i·4-s + (0.0887 − 2.23i)5-s + (0.223 − 1.71i)6-s + (−1.91 − 1.91i)7-s + (−0.707 − 0.707i)8-s + (0.768 − 2.89i)9-s + (−1.51 − 1.64i)10-s + 2.90i·11-s + (−1.05 − 1.37i)12-s + (−3.30 + 3.30i)13-s − 2.70·14-s + (−2.23 − 3.16i)15-s − 1.00·16-s + (1.96 − 1.96i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.792 − 0.609i)3-s − 0.500i·4-s + (0.0396 − 0.999i)5-s + (0.0913 − 0.701i)6-s + (−0.724 − 0.724i)7-s + (−0.250 − 0.250i)8-s + (0.256 − 0.966i)9-s + (−0.479 − 0.519i)10-s + 0.874i·11-s + (−0.304 − 0.396i)12-s + (−0.916 + 0.916i)13-s − 0.724·14-s + (−0.577 − 0.816i)15-s − 0.250·16-s + (0.476 − 0.476i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 + 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.927 + 0.374i$
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: \((2,\ 930,\ (\ :1/2),\ -0.927 + 0.374i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.428345 - 2.20169i\)
\(L(\frac12)\) \(\approx\) \(0.428345 - 2.20169i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.37 + 1.05i)T \)
5 \( 1 + (-0.0887 + 2.23i)T \)
31 \( 1 - T \)
good7 \( 1 + (1.91 + 1.91i)T + 7iT^{2} \)
11 \( 1 - 2.90iT - 11T^{2} \)
13 \( 1 + (3.30 - 3.30i)T - 13iT^{2} \)
17 \( 1 + (-1.96 + 1.96i)T - 17iT^{2} \)
19 \( 1 + 4.24iT - 19T^{2} \)
23 \( 1 + (-1.11 - 1.11i)T + 23iT^{2} \)
29 \( 1 - 8.88T + 29T^{2} \)
37 \( 1 + (-2.16 - 2.16i)T + 37iT^{2} \)
41 \( 1 - 3.64iT - 41T^{2} \)
43 \( 1 + (-1.90 + 1.90i)T - 43iT^{2} \)
47 \( 1 + (2.85 - 2.85i)T - 47iT^{2} \)
53 \( 1 + (2.41 + 2.41i)T + 53iT^{2} \)
59 \( 1 - 14.8T + 59T^{2} \)
61 \( 1 + 6.35T + 61T^{2} \)
67 \( 1 + (0.273 + 0.273i)T + 67iT^{2} \)
71 \( 1 + 12.0iT - 71T^{2} \)
73 \( 1 + (-4.09 + 4.09i)T - 73iT^{2} \)
79 \( 1 + 17.5iT - 79T^{2} \)
83 \( 1 + (-0.573 - 0.573i)T + 83iT^{2} \)
89 \( 1 - 1.57T + 89T^{2} \)
97 \( 1 + (-5.85 - 5.85i)T + 97iT^{2} \)
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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

−9.551397543417068978560501635356, −9.179156494210272368039055194807, −7.979611952040861024129615151903, −7.08102253063006062625757101292, −6.45878371728986855384899980503, −4.91699341610497394896240228426, −4.34121571962601802506826784964, −3.14795292112323911761996602398, −2.08348150982303810515646598051, −0.810679468934393529984734309462, 2.58477980070499282210205877988, 3.08450162125636983662141909923, 3.99119035384312737800382322614, 5.35514449744177160014025155471, 6.02179018052471581379656199967, 7.02161327176178237846615144900, 8.008441793270991581573878623065, 8.540686145135200084425801451500, 9.747817928700196654243300297112, 10.19960157902997347948057631070

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