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Properties

Label 2-930-15.8-c1-0-1
Degree $2$
Conductor $930$
Sign $-0.806 + 0.590i$
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 + (−0.673 + 1.59i)3-s + 1.00i·4-s + (−1.55 + 1.60i)5-s + (1.60 − 0.652i)6-s + (−0.700 + 0.700i)7-s + (0.707 − 0.707i)8-s + (−2.09 − 2.14i)9-s + (2.23 − 0.0306i)10-s + 4.42i·11-s + (−1.59 − 0.673i)12-s + (0.282 + 0.282i)13-s + 0.990·14-s + (−1.50 − 3.56i)15-s − 1.00·16-s + (−1.25 − 1.25i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.388 + 0.921i)3-s + 0.500i·4-s + (−0.697 + 0.716i)5-s + (0.655 − 0.266i)6-s + (−0.264 + 0.264i)7-s + (0.250 − 0.250i)8-s + (−0.697 − 0.716i)9-s + (0.707 − 0.00969i)10-s + 1.33i·11-s + (−0.460 − 0.194i)12-s + (0.0783 + 0.0783i)13-s + 0.264·14-s + (−0.389 − 0.921i)15-s − 0.250·16-s + (−0.305 − 0.305i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.806 + 0.590i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.806 + 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.806 + 0.590i$
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: \((2,\ 930,\ (\ :1/2),\ -0.806 + 0.590i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0685702 - 0.209807i\)
\(L(\frac12)\) \(\approx\) \(0.0685702 - 0.209807i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.673 - 1.59i)T \)
5 \( 1 + (1.55 - 1.60i)T \)
31 \( 1 - T \)
good7 \( 1 + (0.700 - 0.700i)T - 7iT^{2} \)
11 \( 1 - 4.42iT - 11T^{2} \)
13 \( 1 + (-0.282 - 0.282i)T + 13iT^{2} \)
17 \( 1 + (1.25 + 1.25i)T + 17iT^{2} \)
19 \( 1 - 5.10iT - 19T^{2} \)
23 \( 1 + (3.17 - 3.17i)T - 23iT^{2} \)
29 \( 1 - 0.862T + 29T^{2} \)
37 \( 1 + (1.12 - 1.12i)T - 37iT^{2} \)
41 \( 1 + 9.40iT - 41T^{2} \)
43 \( 1 + (5.61 + 5.61i)T + 43iT^{2} \)
47 \( 1 + (5.96 + 5.96i)T + 47iT^{2} \)
53 \( 1 + (-7.34 + 7.34i)T - 53iT^{2} \)
59 \( 1 - 0.852T + 59T^{2} \)
61 \( 1 + 3.32T + 61T^{2} \)
67 \( 1 + (-7.00 + 7.00i)T - 67iT^{2} \)
71 \( 1 - 2.54iT - 71T^{2} \)
73 \( 1 + (8.92 + 8.92i)T + 73iT^{2} \)
79 \( 1 - 2.12iT - 79T^{2} \)
83 \( 1 + (-3.12 + 3.12i)T - 83iT^{2} \)
89 \( 1 - 2.60T + 89T^{2} \)
97 \( 1 + (11.2 - 11.2i)T - 97iT^{2} \)
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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

−10.29303830993810147545005880333, −10.05471953505744215950648727318, −9.133701461698518864171736706732, −8.194292874850889345863478919537, −7.25583878777739672039711111820, −6.39859817369023905229647504710, −5.17520050186333998366644942436, −4.07634518080364501589966029268, −3.43560387457385420687117391706, −2.13914946113032762855442682653, 0.14355555230545557804444244883, 1.19232339080186031572068904148, 2.90547278793004025091049431407, 4.38046765394220639299843079257, 5.41518063216178136909987472784, 6.31584059490612980508956434011, 6.99108237792556863316067975795, 8.100986616704972216249712942876, 8.363366518852579234700245704128, 9.275809916512784367849538085453

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