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Properties

Label 2-930-15.2-c1-0-30
Degree $2$
Conductor $930$
Sign $0.990 - 0.140i$
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 + (1.59 + 0.673i)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 + (0.673 − 1.59i)12-s + (0.282 − 0.282i)13-s − 0.990·14-s + (1.40 + 3.60i)15-s − 1.00·16-s + (1.25 − 1.25i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.921 + 0.388i)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 + (0.194 − 0.460i)12-s + (0.0783 − 0.0783i)13-s − 0.264·14-s + (0.363 + 0.931i)15-s − 0.250·16-s + (0.305 − 0.305i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.140i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.990 - 0.140i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (497, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.990 - 0.140i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.09363 + 0.217995i\)
\(L(\frac12)\) \(\approx\) \(3.09363 + 0.217995i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.59 - 0.673i)T \)
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

−9.930689515598453471223393973980, −9.625636549570520208614194255152, −8.679532304798380801575636488777, −7.21550039967988563165891595848, −6.96522873219627245246114214425, −5.48069785474926611791670416450, −4.63329700930252677786472525812, −3.55255106614923740631042233192, −2.69715749801656804077770091531, −1.78389719445113728669972796827, 1.32054067528193721629189323031, 2.73325794132892970803593814024, 3.62753455090804047031174161185, 4.78469174755245294382132974240, 5.97980903776200434208125712575, 6.33806805029762310139727909859, 7.67901003617816217102410465093, 8.381048200116650311156597764560, 8.959091289868166379427547315397, 9.758852558490959232177921686468

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