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Properties

Label 2-930-15.2-c1-0-23
Degree $2$
Conductor $930$
Sign $0.374 - 0.927i$
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 + (1.70 − 0.292i)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 + 4.24i·11-s + (−0.292 − 1.70i)12-s + (−3 + 3i)13-s + (3.82 + 0.585i)15-s − 1.00·16-s + (−1.41 + 1.41i)17-s + (−1.29 + 2.70i)18-s + 4i·19-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.985 − 0.169i)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 + 1.27i·11-s + (−0.0845 − 0.492i)12-s + (−0.832 + 0.832i)13-s + (0.988 + 0.151i)15-s − 0.250·16-s + (−0.342 + 0.342i)17-s + (−0.304 + 0.638i)18-s + 0.917i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 - 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.374 - 0.927i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (497, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.374 - 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.63688 + 1.10446i\)
\(L(\frac12)\) \(\approx\) \(1.63688 + 1.10446i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.70 + 0.292i)T \)
5 \( 1 + (-2.12 - 0.707i)T \)
31 \( 1 + T \)
good7 \( 1 + 7iT^{2} \)
11 \( 1 - 4.24iT - 11T^{2} \)
13 \( 1 + (3 - 3i)T - 13iT^{2} \)
17 \( 1 + (1.41 - 1.41i)T - 17iT^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + 23iT^{2} \)
29 \( 1 - 2.82T + 29T^{2} \)
37 \( 1 + (5 + 5i)T + 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (-8 + 8i)T - 43iT^{2} \)
47 \( 1 + (-4.24 + 4.24i)T - 47iT^{2} \)
53 \( 1 + (-4.24 - 4.24i)T + 53iT^{2} \)
59 \( 1 - 5.65T + 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 + (1 + i)T + 67iT^{2} \)
71 \( 1 + 1.41iT - 71T^{2} \)
73 \( 1 + (-2 + 2i)T - 73iT^{2} \)
79 \( 1 + 10iT - 79T^{2} \)
83 \( 1 + (12.7 + 12.7i)T + 83iT^{2} \)
89 \( 1 - 7.07T + 89T^{2} \)
97 \( 1 + (-3 - 3i)T + 97iT^{2} \)
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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

−10.13489668482817216505887647601, −9.220340357206454379453383100712, −8.786988460348386542965585782847, −7.48627357674273404705534502216, −7.11733374346453031684230475801, −6.20193679202515745442359201255, −5.01503037967307435874493270303, −3.94653290967398345098659463652, −2.34924541118969268871658147381, −1.77191542487634168713594702616, 1.03492709954164932533007146838, 2.51449263104060758167431677344, 3.02860342749846366064179571852, 4.44082368979591677309874429857, 5.43590190153375436318456036634, 6.65067766531095354132003118116, 7.68046659576004719456470652217, 8.525229944155995200867241237113, 9.095846065782067224911676917412, 9.796352728833768947102373890726

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