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Properties

Label 2-930-15.8-c1-0-28
Degree $2$
Conductor $930$
Sign $0.749 - 0.662i$
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 + (0.292 + 1.70i)3-s + 1.00i·4-s + (2.12 − 0.707i)5-s + (0.999 − 1.41i)6-s + (0.707 − 0.707i)8-s + (−2.82 + i)9-s + (−2 − 0.999i)10-s − 1.41i·11-s + (−1.70 + 0.292i)12-s + (3 + 3i)13-s + (1.82 + 3.41i)15-s − 1.00·16-s + (1.41 + 1.41i)17-s + (2.70 + 1.29i)18-s + 4i·19-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (0.169 + 0.985i)3-s + 0.500i·4-s + (0.948 − 0.316i)5-s + (0.408 − 0.577i)6-s + (0.250 − 0.250i)8-s + (−0.942 + 0.333i)9-s + (−0.632 − 0.316i)10-s − 0.426i·11-s + (−0.492 + 0.0845i)12-s + (0.832 + 0.832i)13-s + (0.472 + 0.881i)15-s − 0.250·16-s + (0.342 + 0.342i)17-s + (0.638 + 0.304i)18-s + 0.917i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.749 - 0.662i$
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: \((2,\ 930,\ (\ :1/2),\ 0.749 - 0.662i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43114 + 0.541543i\)
\(L(\frac12)\) \(\approx\) \(1.43114 + 0.541543i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.292 - 1.70i)T \)
5 \( 1 + (-2.12 + 0.707i)T \)
31 \( 1 - T \)
good7 \( 1 - 7iT^{2} \)
11 \( 1 + 1.41iT - 11T^{2} \)
13 \( 1 + (-3 - 3i)T + 13iT^{2} \)
17 \( 1 + (-1.41 - 1.41i)T + 17iT^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + (-2.82 + 2.82i)T - 23iT^{2} \)
29 \( 1 - 2.82T + 29T^{2} \)
37 \( 1 + (3 - 3i)T - 37iT^{2} \)
41 \( 1 - 5.65iT - 41T^{2} \)
43 \( 1 + 43iT^{2} \)
47 \( 1 + (1.41 + 1.41i)T + 47iT^{2} \)
53 \( 1 + (1.41 - 1.41i)T - 53iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 + (-7 + 7i)T - 67iT^{2} \)
71 \( 1 - 12.7iT - 71T^{2} \)
73 \( 1 + (2 + 2i)T + 73iT^{2} \)
79 \( 1 + 2iT - 79T^{2} \)
83 \( 1 + (4.24 - 4.24i)T - 83iT^{2} \)
89 \( 1 + 7.07T + 89T^{2} \)
97 \( 1 + (-11 + 11i)T - 97iT^{2} \)
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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

−10.08559065732896922231829249635, −9.457075548651978693647896275854, −8.658997787202895002106483943066, −8.190920736289732949377917388440, −6.60118200056126273517061255496, −5.78827381665689402033656916842, −4.73485587314305410390142953187, −3.75142164105122242155079255180, −2.70313833160141553854625577755, −1.39013957921369484586671644610, 0.954639368283717911635974826442, 2.16753496969179128945682982213, 3.25524980856768513062933791808, 5.15553263580872571377333604639, 5.82998894297209056577540208367, 6.76147903473644709712080002943, 7.26891475208656824449279522470, 8.299368091213459864830191772305, 8.993428667336132716539890471810, 9.811881701617139466437841959489

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