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Properties

Label 2-930-15.8-c1-0-8
Degree $2$
Conductor $930$
Sign $0.460 - 0.887i$
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 + (−1.50 − 0.860i)3-s + 1.00i·4-s + (2.19 + 0.424i)5-s + (0.454 + 1.67i)6-s + (−0.358 + 0.358i)7-s + (0.707 − 0.707i)8-s + (1.52 + 2.58i)9-s + (−1.25 − 1.85i)10-s + 1.95i·11-s + (0.860 − 1.50i)12-s + (1.74 + 1.74i)13-s + 0.507·14-s + (−2.93 − 2.52i)15-s − 1.00·16-s + (−3.65 − 3.65i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.867 − 0.496i)3-s + 0.500i·4-s + (0.981 + 0.189i)5-s + (0.185 + 0.682i)6-s + (−0.135 + 0.135i)7-s + (0.250 − 0.250i)8-s + (0.506 + 0.862i)9-s + (−0.395 − 0.585i)10-s + 0.589i·11-s + (0.248 − 0.433i)12-s + (0.482 + 0.482i)13-s + 0.135·14-s + (−0.757 − 0.652i)15-s − 0.250·16-s + (−0.887 − 0.887i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.460 - 0.887i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.460 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.460 - 0.887i$
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: \((2,\ 930,\ (\ :1/2),\ 0.460 - 0.887i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.605115 + 0.367631i\)
\(L(\frac12)\) \(\approx\) \(0.605115 + 0.367631i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.50 + 0.860i)T \)
5 \( 1 + (-2.19 - 0.424i)T \)
31 \( 1 + T \)
good7 \( 1 + (0.358 - 0.358i)T - 7iT^{2} \)
11 \( 1 - 1.95iT - 11T^{2} \)
13 \( 1 + (-1.74 - 1.74i)T + 13iT^{2} \)
17 \( 1 + (3.65 + 3.65i)T + 17iT^{2} \)
19 \( 1 - 3.69iT - 19T^{2} \)
23 \( 1 + (6.27 - 6.27i)T - 23iT^{2} \)
29 \( 1 + 8.51T + 29T^{2} \)
37 \( 1 + (1.92 - 1.92i)T - 37iT^{2} \)
41 \( 1 - 9.77iT - 41T^{2} \)
43 \( 1 + (1.51 + 1.51i)T + 43iT^{2} \)
47 \( 1 + (4.14 + 4.14i)T + 47iT^{2} \)
53 \( 1 + (-9.34 + 9.34i)T - 53iT^{2} \)
59 \( 1 + 1.87T + 59T^{2} \)
61 \( 1 - 10.5T + 61T^{2} \)
67 \( 1 + (7.86 - 7.86i)T - 67iT^{2} \)
71 \( 1 - 2.88iT - 71T^{2} \)
73 \( 1 + (-9.70 - 9.70i)T + 73iT^{2} \)
79 \( 1 - 1.35iT - 79T^{2} \)
83 \( 1 + (2.46 - 2.46i)T - 83iT^{2} \)
89 \( 1 - 0.192T + 89T^{2} \)
97 \( 1 + (-1.01 + 1.01i)T - 97iT^{2} \)
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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

−9.939140931171623147518808971734, −9.819072207425304613933861089777, −8.686401651865709013319477405014, −7.56672559598044507244305651120, −6.81752453324708270031760783052, −5.97547267965618262356094495030, −5.13030275588907534070196397828, −3.85269954821749415421421034998, −2.25038402666189816849265210147, −1.53614488037589229228570112334, 0.43498682487927415188297241579, 2.01978262406976055107578110881, 3.78546401702181367143358528488, 4.88240630404748525627322478618, 5.88556609490340507987939728327, 6.20651246139562099516337767351, 7.18626576395375966129353317755, 8.540693344987936661169474295001, 9.050563906773247330142438018899, 9.989956863377883671990559418588

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