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Properties

Label 2-930-15.8-c1-0-31
Degree $2$
Conductor $930$
Sign $-0.0239 - 0.999i$
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 + (1.68 − 0.404i)3-s + 1.00i·4-s + (2.01 + 0.959i)5-s + (1.47 + 0.904i)6-s + (−2.95 + 2.95i)7-s + (−0.707 + 0.707i)8-s + (2.67 − 1.36i)9-s + (0.749 + 2.10i)10-s + 4.99i·11-s + (0.404 + 1.68i)12-s + (−0.576 − 0.576i)13-s − 4.18·14-s + (3.78 + 0.799i)15-s − 1.00·16-s + (−1.92 − 1.92i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.972 − 0.233i)3-s + 0.500i·4-s + (0.903 + 0.429i)5-s + (0.602 + 0.369i)6-s + (−1.11 + 1.11i)7-s + (−0.250 + 0.250i)8-s + (0.890 − 0.454i)9-s + (0.236 + 0.666i)10-s + 1.50i·11-s + (0.116 + 0.486i)12-s + (−0.159 − 0.159i)13-s − 1.11·14-s + (0.978 + 0.206i)15-s − 0.250·16-s + (−0.467 − 0.467i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0239 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0239 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.0239 - 0.999i$
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: \((2,\ 930,\ (\ :1/2),\ -0.0239 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.02256 + 2.07158i\)
\(L(\frac12)\) \(\approx\) \(2.02256 + 2.07158i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.68 + 0.404i)T \)
5 \( 1 + (-2.01 - 0.959i)T \)
31 \( 1 + T \)
good7 \( 1 + (2.95 - 2.95i)T - 7iT^{2} \)
11 \( 1 - 4.99iT - 11T^{2} \)
13 \( 1 + (0.576 + 0.576i)T + 13iT^{2} \)
17 \( 1 + (1.92 + 1.92i)T + 17iT^{2} \)
19 \( 1 + 2.28iT - 19T^{2} \)
23 \( 1 + (-2.35 + 2.35i)T - 23iT^{2} \)
29 \( 1 + 7.87T + 29T^{2} \)
37 \( 1 + (-5.67 + 5.67i)T - 37iT^{2} \)
41 \( 1 + 6.89iT - 41T^{2} \)
43 \( 1 + (-7.51 - 7.51i)T + 43iT^{2} \)
47 \( 1 + (-4.39 - 4.39i)T + 47iT^{2} \)
53 \( 1 + (-3.78 + 3.78i)T - 53iT^{2} \)
59 \( 1 - 8.23T + 59T^{2} \)
61 \( 1 - 2.57T + 61T^{2} \)
67 \( 1 + (-8.10 + 8.10i)T - 67iT^{2} \)
71 \( 1 - 0.808iT - 71T^{2} \)
73 \( 1 + (-6.98 - 6.98i)T + 73iT^{2} \)
79 \( 1 + 6.04iT - 79T^{2} \)
83 \( 1 + (5.46 - 5.46i)T - 83iT^{2} \)
89 \( 1 + 13.0T + 89T^{2} \)
97 \( 1 + (-9.00 + 9.00i)T - 97iT^{2} \)
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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

−9.771269481507185677608196813054, −9.432670849197105614280510326144, −8.799904934945121847414303213041, −7.42152105057046761545497542820, −6.94128569979023536757834303287, −6.09690531502497224070731258813, −5.14009328728176397174127092093, −3.91446189267061604701871041770, −2.62192716102275505956110518347, −2.29007143575707735666625175575, 1.08029058902690369282257554238, 2.47776498951456318965874936922, 3.51394454798393515610654446960, 4.09970605514757414415163859719, 5.44001483530481367085691354932, 6.30082278284902142565200607690, 7.25154029712816976234868142842, 8.439757713736295982350579477031, 9.229120201095454480639046218630, 9.859227673047734346689596393833

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