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Properties

Label 2-930-155.37-c1-0-12
Degree $2$
Conductor $930$
Sign $0.623 - 0.781i$
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 + (−0.258 + 0.965i)3-s + 1.00i·4-s + (1.64 + 1.51i)5-s + (0.866 − 0.500i)6-s + (−0.191 + 0.713i)7-s + (0.707 − 0.707i)8-s + (−0.866 − 0.499i)9-s + (−0.0966 − 2.23i)10-s + (−0.353 − 0.204i)11-s + (−0.965 − 0.258i)12-s + (3.20 − 0.857i)13-s + (0.640 − 0.369i)14-s + (−1.88 + 1.20i)15-s − 1.00·16-s + (6.70 + 1.79i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.149 + 0.557i)3-s + 0.500i·4-s + (0.737 + 0.675i)5-s + (0.353 − 0.204i)6-s + (−0.0723 + 0.269i)7-s + (0.250 − 0.250i)8-s + (−0.288 − 0.166i)9-s + (−0.0305 − 0.706i)10-s + (−0.106 − 0.0615i)11-s + (−0.278 − 0.0747i)12-s + (0.887 − 0.237i)13-s + (0.171 − 0.0987i)14-s + (−0.487 + 0.310i)15-s − 0.250·16-s + (1.62 + 0.435i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.623 - 0.781i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.623 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.623 - 0.781i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.623 - 0.781i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21413 + 0.584856i\)
\(L(\frac12)\) \(\approx\) \(1.21413 + 0.584856i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.258 - 0.965i)T \)
5 \( 1 + (-1.64 - 1.51i)T \)
31 \( 1 + (1.13 - 5.45i)T \)
good7 \( 1 + (0.191 - 0.713i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (0.353 + 0.204i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.20 + 0.857i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-6.70 - 1.79i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.40 + 0.812i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.33 + 4.33i)T + 23iT^{2} \)
29 \( 1 - 9.18T + 29T^{2} \)
37 \( 1 + (5.87 + 1.57i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (3.23 - 5.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.01 - 11.2i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (3.91 + 3.91i)T + 47iT^{2} \)
53 \( 1 + (11.0 - 2.97i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-9.86 + 5.69i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 5.13iT - 61T^{2} \)
67 \( 1 + (0.811 - 0.217i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-0.954 + 1.65i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (10.2 - 2.74i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-3.08 - 5.34i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-12.5 + 3.35i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 - 7.13T + 89T^{2} \)
97 \( 1 + (0.978 + 0.978i)T + 97iT^{2} \)
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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

−10.26685843830294726007218795086, −9.633182780464756387775962835087, −8.620683345293137873288577280672, −7.954399416179557085287328683133, −6.64094894381236883597950318402, −5.94733520801530614632660628728, −4.91665160006541452402330882357, −3.51069899076426173403756973221, −2.85222759758483484175962547258, −1.38217386000614918580156888715, 0.881613481638484263849368388581, 1.90883039936552864233584287964, 3.56223926304298536056834302793, 5.05096206310832411437840194444, 5.73716833732919760076072571705, 6.50343255508876284144008791075, 7.50362727555622596644048382625, 8.231686635823437940131559651804, 9.003364498739027534718593515004, 9.953434666344638833705467092872

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