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Properties

Label 2-930-155.37-c1-0-18
Degree $2$
Conductor $930$
Sign $0.266 + 0.963i$
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 + (−1.98 + 1.03i)5-s + (0.866 − 0.500i)6-s + (0.211 − 0.790i)7-s + (0.707 − 0.707i)8-s + (−0.866 − 0.499i)9-s + (2.13 + 0.669i)10-s + (−0.702 − 0.405i)11-s + (−0.965 − 0.258i)12-s + (−4.35 + 1.16i)13-s + (−0.708 + 0.409i)14-s + (−0.486 − 2.18i)15-s − 1.00·16-s + (0.689 + 0.184i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.149 + 0.557i)3-s + 0.500i·4-s + (−0.886 + 0.462i)5-s + (0.353 − 0.204i)6-s + (0.0800 − 0.298i)7-s + (0.250 − 0.250i)8-s + (−0.288 − 0.166i)9-s + (0.674 + 0.211i)10-s + (−0.211 − 0.122i)11-s + (−0.278 − 0.0747i)12-s + (−1.20 + 0.323i)13-s + (−0.189 + 0.109i)14-s + (−0.125 − 0.563i)15-s − 0.250·16-s + (0.167 + 0.0447i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.266 + 0.963i$
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: \((2,\ 930,\ (\ :1/2),\ 0.266 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.506666 - 0.385736i\)
\(L(\frac12)\) \(\approx\) \(0.506666 - 0.385736i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.98 - 1.03i)T \)
31 \( 1 + (2.60 - 4.92i)T \)
good7 \( 1 + (-0.211 + 0.790i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (0.702 + 0.405i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.35 - 1.16i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-0.689 - 0.184i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-6.50 + 3.75i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.45 + 3.45i)T + 23iT^{2} \)
29 \( 1 + 2.47T + 29T^{2} \)
37 \( 1 + (-10.2 - 2.75i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-3.26 + 5.66i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.54 + 9.51i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (1.76 + 1.76i)T + 47iT^{2} \)
53 \( 1 + (9.01 - 2.41i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-10.3 + 5.96i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 13.7iT - 61T^{2} \)
67 \( 1 + (-12.2 + 3.27i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-4.94 + 8.56i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.17 + 0.314i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (0.836 + 1.44i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (15.6 - 4.18i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + 4.90T + 89T^{2} \)
97 \( 1 + (9.66 + 9.66i)T + 97iT^{2} \)
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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

−9.887566624883619669966506744376, −9.319268148859823616003316949250, −8.218006098673772617455304005411, −7.47955856703376450578211546365, −6.78400493423118601386244136360, −5.28645823809803313731245578153, −4.37383278397408814951478082508, −3.43794560420331060854947109410, −2.45844237444158077374647634513, −0.43169183120054488197863634317, 1.09726373931068502771795064073, 2.64039997666382644684213842087, 4.09692012526500427567639924607, 5.27451782002229571091915470041, 5.86496197364553305230807995631, 7.26635886437797968818208212397, 7.68375180538560363471571102522, 8.225427143960060941458651745823, 9.483721224009263369111450464083, 9.871293393128246205930379274880

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