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Properties

Label 2-930-15.8-c1-0-19
Degree $2$
Conductor $930$
Sign $-0.978 - 0.205i$
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 + (0.00608 + 1.73i)3-s + 1.00i·4-s + (2.02 + 0.956i)5-s + (−1.22 + 1.22i)6-s + (−1.11 + 1.11i)7-s + (−0.707 + 0.707i)8-s + (−2.99 + 0.0210i)9-s + (0.753 + 2.10i)10-s + 2.70i·11-s + (−1.73 + 0.00608i)12-s + (0.120 + 0.120i)13-s − 1.57·14-s + (−1.64 + 3.50i)15-s − 1.00·16-s + (−0.852 − 0.852i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.00351 + 0.999i)3-s + 0.500i·4-s + (0.903 + 0.427i)5-s + (−0.498 + 0.501i)6-s + (−0.419 + 0.419i)7-s + (−0.250 + 0.250i)8-s + (−0.999 + 0.00703i)9-s + (0.238 + 0.665i)10-s + 0.816i·11-s + (−0.499 + 0.00175i)12-s + (0.0334 + 0.0334i)13-s − 0.419·14-s + (−0.424 + 0.905i)15-s − 0.250·16-s + (−0.206 − 0.206i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.205i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 - 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.978 - 0.205i$
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: \((2,\ 930,\ (\ :1/2),\ -0.978 - 0.205i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.204694 + 1.97527i\)
\(L(\frac12)\) \(\approx\) \(0.204694 + 1.97527i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.00608 - 1.73i)T \)
5 \( 1 + (-2.02 - 0.956i)T \)
31 \( 1 - T \)
good7 \( 1 + (1.11 - 1.11i)T - 7iT^{2} \)
11 \( 1 - 2.70iT - 11T^{2} \)
13 \( 1 + (-0.120 - 0.120i)T + 13iT^{2} \)
17 \( 1 + (0.852 + 0.852i)T + 17iT^{2} \)
19 \( 1 + 5.63iT - 19T^{2} \)
23 \( 1 + (6.00 - 6.00i)T - 23iT^{2} \)
29 \( 1 - 9.37T + 29T^{2} \)
37 \( 1 + (-4.37 + 4.37i)T - 37iT^{2} \)
41 \( 1 - 12.3iT - 41T^{2} \)
43 \( 1 + (6.77 + 6.77i)T + 43iT^{2} \)
47 \( 1 + (-1.75 - 1.75i)T + 47iT^{2} \)
53 \( 1 + (0.0579 - 0.0579i)T - 53iT^{2} \)
59 \( 1 + 1.06T + 59T^{2} \)
61 \( 1 - 7.56T + 61T^{2} \)
67 \( 1 + (-5.10 + 5.10i)T - 67iT^{2} \)
71 \( 1 + 15.7iT - 71T^{2} \)
73 \( 1 + (3.90 + 3.90i)T + 73iT^{2} \)
79 \( 1 - 7.85iT - 79T^{2} \)
83 \( 1 + (8.60 - 8.60i)T - 83iT^{2} \)
89 \( 1 - 5.43T + 89T^{2} \)
97 \( 1 + (2.97 - 2.97i)T - 97iT^{2} \)
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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

−10.23117483955004217656191742962, −9.616770584414391510543963993540, −9.026696003183559300260958339675, −7.921619700311979493691488602428, −6.74565232653385581403228662426, −6.10403219339838768942351970752, −5.18366509059196082410228071175, −4.45068511610904436260004057563, −3.20391567388182181947838531306, −2.34860464734828547509911302753, 0.77910705589038721755025498891, 1.96774789836153288353282563693, 2.98601924155163750888239583279, 4.22067047565458558643707889866, 5.50717432272369260663230265789, 6.17907430261283006189925785166, 6.76994813608700178567489344152, 8.229247057014724240697493628728, 8.643682620296929931742186131149, 10.03735074528365385939307692003

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