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Properties

Label 2-930-155.37-c1-0-4
Degree $2$
Conductor $930$
Sign $-0.265 - 0.964i$
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 + (−1.08 − 1.95i)5-s + (0.866 − 0.500i)6-s + (0.294 − 1.09i)7-s + (0.707 − 0.707i)8-s + (−0.866 − 0.499i)9-s + (−0.610 + 2.15i)10-s + (−0.779 − 0.449i)11-s + (−0.965 − 0.258i)12-s + (−4.95 + 1.32i)13-s + (−0.985 + 0.569i)14-s + (2.16 − 0.547i)15-s − 1.00·16-s + (3.02 + 0.810i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.149 + 0.557i)3-s + 0.500i·4-s + (−0.487 − 0.873i)5-s + (0.353 − 0.204i)6-s + (0.111 − 0.415i)7-s + (0.250 − 0.250i)8-s + (−0.288 − 0.166i)9-s + (−0.192 + 0.680i)10-s + (−0.234 − 0.135i)11-s + (−0.278 − 0.0747i)12-s + (−1.37 + 0.368i)13-s + (−0.263 + 0.152i)14-s + (0.559 − 0.141i)15-s − 0.250·16-s + (0.733 + 0.196i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.265 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.265 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.265 - 0.964i$
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: \((2,\ 930,\ (\ :1/2),\ -0.265 - 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.208575 + 0.273886i\)
\(L(\frac12)\) \(\approx\) \(0.208575 + 0.273886i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.08 + 1.95i)T \)
31 \( 1 + (4.84 + 2.74i)T \)
good7 \( 1 + (-0.294 + 1.09i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (0.779 + 0.449i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.95 - 1.32i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-3.02 - 0.810i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (2.07 - 1.19i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.46 - 5.46i)T + 23iT^{2} \)
29 \( 1 + 0.504T + 29T^{2} \)
37 \( 1 + (3.75 + 1.00i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-3.00 + 5.21i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.90 - 10.8i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-7.21 - 7.21i)T + 47iT^{2} \)
53 \( 1 + (13.0 - 3.48i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (9.82 - 5.67i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 13.7iT - 61T^{2} \)
67 \( 1 + (-0.879 + 0.235i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-4.82 + 8.35i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (13.5 - 3.64i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (5.90 + 10.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.04 + 0.548i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + 15.1T + 89T^{2} \)
97 \( 1 + (-10.5 - 10.5i)T + 97iT^{2} \)
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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

−10.30859825757706939957248812465, −9.353908512493124228986042360459, −9.011888777729088077713214995346, −7.71493387027081298421317843543, −7.42402610258509194801046317208, −5.78658425110987198854169916046, −4.81862065932571721333413947735, −4.08381961610043022091812648530, −2.97264586581744233398564157753, −1.37916300724968987419311004736, 0.20008563624151559282648584553, 2.15344458959785163964802891727, 3.13493285309375471211076364826, 4.77494822486135917290470358330, 5.59858142960824550984703583927, 6.78152239776066127111112122565, 7.17909593793743985762850318493, 7.996539713782858312252448340836, 8.795620334506366514763161721900, 9.865570695769748708288354874122

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