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Properties

Label 2-930-15.8-c1-0-37
Degree $2$
Conductor $930$
Sign $0.999 - 0.0286i$
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 + (0.292 − 1.70i)3-s + 1.00i·4-s + (2.23 + 0.0743i)5-s + (1.41 − 0.999i)6-s + (0.218 − 0.218i)7-s + (−0.707 + 0.707i)8-s + (−2.82 − i)9-s + (1.52 + 1.63i)10-s + 0.894i·11-s + (1.70 + 0.292i)12-s + (3.57 + 3.57i)13-s + 0.309·14-s + (0.781 − 3.79i)15-s − 1.00·16-s + (4.36 + 4.36i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.169 − 0.985i)3-s + 0.500i·4-s + (0.999 + 0.0332i)5-s + (0.577 − 0.408i)6-s + (0.0826 − 0.0826i)7-s + (−0.250 + 0.250i)8-s + (−0.942 − 0.333i)9-s + (0.483 + 0.516i)10-s + 0.269i·11-s + (0.492 + 0.0845i)12-s + (0.991 + 0.991i)13-s + 0.0826·14-s + (0.201 − 0.979i)15-s − 0.250·16-s + (1.05 + 1.05i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0286i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0286i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.999 - 0.0286i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (683, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.999 - 0.0286i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.65877 + 0.0381167i\)
\(L(\frac12)\) \(\approx\) \(2.65877 + 0.0381167i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.292 + 1.70i)T \)
5 \( 1 + (-2.23 - 0.0743i)T \)
31 \( 1 - T \)
good7 \( 1 + (-0.218 + 0.218i)T - 7iT^{2} \)
11 \( 1 - 0.894iT - 11T^{2} \)
13 \( 1 + (-3.57 - 3.57i)T + 13iT^{2} \)
17 \( 1 + (-4.36 - 4.36i)T + 17iT^{2} \)
19 \( 1 + 6.55iT - 19T^{2} \)
23 \( 1 + (-3.19 + 3.19i)T - 23iT^{2} \)
29 \( 1 + 2.39T + 29T^{2} \)
37 \( 1 + (2 - 2i)T - 37iT^{2} \)
41 \( 1 + 4.71iT - 41T^{2} \)
43 \( 1 + (3.37 + 3.37i)T + 43iT^{2} \)
47 \( 1 + (8.98 + 8.98i)T + 47iT^{2} \)
53 \( 1 + (-10.1 + 10.1i)T - 53iT^{2} \)
59 \( 1 - 6.79T + 59T^{2} \)
61 \( 1 + 5.61T + 61T^{2} \)
67 \( 1 + (9.52 - 9.52i)T - 67iT^{2} \)
71 \( 1 - 12.5iT - 71T^{2} \)
73 \( 1 + (7.83 + 7.83i)T + 73iT^{2} \)
79 \( 1 - 13.0iT - 79T^{2} \)
83 \( 1 + (-1.55 + 1.55i)T - 83iT^{2} \)
89 \( 1 + 1.83T + 89T^{2} \)
97 \( 1 + (6.81 - 6.81i)T - 97iT^{2} \)
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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

−10.00958428014581297870848967928, −8.824035036845413057326062451015, −8.519475213589854105936000636126, −7.16521872900497256965174317067, −6.70251273710679454683448471523, −5.90381373757271316305751406356, −5.05938708548568955688041185890, −3.70170663543534419359820401990, −2.49357585876319007795012210517, −1.39324981395218617842710575562, 1.38852571434403640442505073796, 2.94639910168199568264377240705, 3.49367260250058818413397624687, 4.82933675081873020040211640785, 5.62280751407268176267886932743, 6.06220104209531018028333049213, 7.68741430156203665509542500044, 8.663514447843153348637002533979, 9.528626689373016073613199805035, 10.09791206847627233463126152930

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