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Properties

Label 2-930-15.8-c1-0-50
Degree $2$
Conductor $930$
Sign $-0.983 + 0.179i$
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 + (0.877 − 1.49i)3-s + 1.00i·4-s + (−1.98 + 1.03i)5-s + (−1.67 + 0.435i)6-s + (2.39 − 2.39i)7-s + (0.707 − 0.707i)8-s + (−1.46 − 2.62i)9-s + (2.13 + 0.673i)10-s − 0.0383i·11-s + (1.49 + 0.877i)12-s + (−2.42 − 2.42i)13-s − 3.38·14-s + (−0.200 + 3.86i)15-s − 1.00·16-s + (3.70 + 3.70i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (0.506 − 0.862i)3-s + 0.500i·4-s + (−0.887 + 0.461i)5-s + (−0.684 + 0.177i)6-s + (0.905 − 0.905i)7-s + (0.250 − 0.250i)8-s + (−0.487 − 0.873i)9-s + (0.674 + 0.213i)10-s − 0.0115i·11-s + (0.431 + 0.253i)12-s + (−0.671 − 0.671i)13-s − 0.905·14-s + (−0.0516 + 0.998i)15-s − 0.250·16-s + (0.898 + 0.898i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.179i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 + 0.179i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.983 + 0.179i$
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: \((2,\ 930,\ (\ :1/2),\ -0.983 + 0.179i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0866316 - 0.959352i\)
\(L(\frac12)\) \(\approx\) \(0.0866316 - 0.959352i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.877 + 1.49i)T \)
5 \( 1 + (1.98 - 1.03i)T \)
31 \( 1 + T \)
good7 \( 1 + (-2.39 + 2.39i)T - 7iT^{2} \)
11 \( 1 + 0.0383iT - 11T^{2} \)
13 \( 1 + (2.42 + 2.42i)T + 13iT^{2} \)
17 \( 1 + (-3.70 - 3.70i)T + 17iT^{2} \)
19 \( 1 + 7.16iT - 19T^{2} \)
23 \( 1 + (3.94 - 3.94i)T - 23iT^{2} \)
29 \( 1 - 3.49T + 29T^{2} \)
37 \( 1 + (4.89 - 4.89i)T - 37iT^{2} \)
41 \( 1 + 10.1iT - 41T^{2} \)
43 \( 1 + (6.93 + 6.93i)T + 43iT^{2} \)
47 \( 1 + (7.35 + 7.35i)T + 47iT^{2} \)
53 \( 1 + (0.287 - 0.287i)T - 53iT^{2} \)
59 \( 1 + 15.0T + 59T^{2} \)
61 \( 1 - 5.16T + 61T^{2} \)
67 \( 1 + (-5.80 + 5.80i)T - 67iT^{2} \)
71 \( 1 + 9.58iT - 71T^{2} \)
73 \( 1 + (-9.96 - 9.96i)T + 73iT^{2} \)
79 \( 1 - 16.7iT - 79T^{2} \)
83 \( 1 + (8.18 - 8.18i)T - 83iT^{2} \)
89 \( 1 - 3.51T + 89T^{2} \)
97 \( 1 + (2.47 - 2.47i)T - 97iT^{2} \)
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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

−9.774124578143287147359246278789, −8.471980627675613789931722743603, −8.077399575563845266203727862830, −7.32444856378820229914218523212, −6.78329729902778906233915722369, −5.15617776272189087761055590417, −3.88824906795741330811007871767, −3.10706679116735051568548812362, −1.83134267502786753108071631895, −0.50387782566110388605074655561, 1.80254029155800050234961956150, 3.21637115405765242472026720139, 4.53960306065007171400044493152, 4.99878259808645430070918240250, 6.07048667751615802464112607588, 7.55951909081244982461460968309, 8.089098804541962622781071862746, 8.632555528728850888687031455853, 9.538116807405343156798001732633, 10.13291529684607073982046160579

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