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Properties

Label 2-930-15.8-c1-0-10
Degree $2$
Conductor $930$
Sign $-0.810 - 0.586i$
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 + (1.33 + 1.10i)3-s + 1.00i·4-s + (−0.368 + 2.20i)5-s + (−0.160 − 1.72i)6-s + (−2.86 + 2.86i)7-s + (0.707 − 0.707i)8-s + (0.553 + 2.94i)9-s + (1.82 − 1.29i)10-s + 0.0722i·11-s + (−1.10 + 1.33i)12-s + (2.91 + 2.91i)13-s + 4.05·14-s + (−2.93 + 2.53i)15-s − 1.00·16-s + (−4.16 − 4.16i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (0.769 + 0.638i)3-s + 0.500i·4-s + (−0.164 + 0.986i)5-s + (−0.0655 − 0.704i)6-s + (−1.08 + 1.08i)7-s + (0.250 − 0.250i)8-s + (0.184 + 0.982i)9-s + (0.575 − 0.410i)10-s + 0.0217i·11-s + (−0.319 + 0.384i)12-s + (0.809 + 0.809i)13-s + 1.08·14-s + (−0.756 + 0.653i)15-s − 0.250·16-s + (−1.01 − 1.01i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.810 - 0.586i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.810 - 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.810 - 0.586i$
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: \((2,\ 930,\ (\ :1/2),\ -0.810 - 0.586i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.304800 + 0.941286i\)
\(L(\frac12)\) \(\approx\) \(0.304800 + 0.941286i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.33 - 1.10i)T \)
5 \( 1 + (0.368 - 2.20i)T \)
31 \( 1 + T \)
good7 \( 1 + (2.86 - 2.86i)T - 7iT^{2} \)
11 \( 1 - 0.0722iT - 11T^{2} \)
13 \( 1 + (-2.91 - 2.91i)T + 13iT^{2} \)
17 \( 1 + (4.16 + 4.16i)T + 17iT^{2} \)
19 \( 1 + 3.12iT - 19T^{2} \)
23 \( 1 + (-1.55 + 1.55i)T - 23iT^{2} \)
29 \( 1 - 3.03T + 29T^{2} \)
37 \( 1 + (5.90 - 5.90i)T - 37iT^{2} \)
41 \( 1 - 1.90iT - 41T^{2} \)
43 \( 1 + (0.388 + 0.388i)T + 43iT^{2} \)
47 \( 1 + (2.00 + 2.00i)T + 47iT^{2} \)
53 \( 1 + (6.68 - 6.68i)T - 53iT^{2} \)
59 \( 1 + 2.33T + 59T^{2} \)
61 \( 1 - 12.8T + 61T^{2} \)
67 \( 1 + (-7.43 + 7.43i)T - 67iT^{2} \)
71 \( 1 + 5.69iT - 71T^{2} \)
73 \( 1 + (-6.06 - 6.06i)T + 73iT^{2} \)
79 \( 1 - 7.62iT - 79T^{2} \)
83 \( 1 + (0.153 - 0.153i)T - 83iT^{2} \)
89 \( 1 + 13.6T + 89T^{2} \)
97 \( 1 + (2.66 - 2.66i)T - 97iT^{2} \)
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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

−10.26188523720504942016852207734, −9.443406402680019934037664280730, −9.010791301439420306306883649531, −8.238038435632689643292479335734, −6.95961320059521457349916120972, −6.44257596595756595697154125583, −4.92206286504624801884962348446, −3.72274508018803023260648398699, −2.90189266199736947617617917970, −2.26486848854394115404800143506, 0.49044967692010230024137255627, 1.65859167189666711054023138804, 3.41719623438191862032945351665, 4.14526859602839389159197492603, 5.67211603890818556359185364437, 6.53331552629357591911498540603, 7.27118001183297962217654533686, 8.232729167907760990302608890301, 8.622270350411316619949515222635, 9.570954668607475999844950616138

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