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Properties

Label 2-930-15.8-c1-0-48
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: $\chi_{930} (683, \cdot )$
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

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

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