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Properties

Label 2-930-5.4-c3-0-60
Degree $2$
Conductor $930$
Sign $0.993 - 0.111i$
Analytic cond. $54.8717$
Root an. cond. $7.40754$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·2-s − 3i·3-s − 4·4-s + (11.1 − 1.24i)5-s + 6·6-s − 14.4i·7-s − 8i·8-s − 9·9-s + (2.49 + 22.2i)10-s + 31.5·11-s + 12i·12-s + 19.1i·13-s + 28.8·14-s + (−3.74 − 33.3i)15-s + 16·16-s + 53.6i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (0.993 − 0.111i)5-s + 0.408·6-s − 0.778i·7-s − 0.353i·8-s − 0.333·9-s + (0.0789 + 0.702i)10-s + 0.866·11-s + 0.288i·12-s + 0.407i·13-s + 0.550·14-s + (−0.0644 − 0.573i)15-s + 0.250·16-s + 0.765i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.111i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.993 - 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.993 - 0.111i$
Analytic conductor: \(54.8717\)
Root analytic conductor: \(7.40754\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :3/2),\ 0.993 - 0.111i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.606201807\)
\(L(\frac12)\) \(\approx\) \(2.606201807\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2iT \)
3 \( 1 + 3iT \)
5 \( 1 + (-11.1 + 1.24i)T \)
31 \( 1 + 31T \)
good7 \( 1 + 14.4iT - 343T^{2} \)
11 \( 1 - 31.5T + 1.33e3T^{2} \)
13 \( 1 - 19.1iT - 2.19e3T^{2} \)
17 \( 1 - 53.6iT - 4.91e3T^{2} \)
19 \( 1 - 51.9T + 6.85e3T^{2} \)
23 \( 1 - 71.5iT - 1.21e4T^{2} \)
29 \( 1 - 63.2T + 2.43e4T^{2} \)
37 \( 1 - 73.2iT - 5.06e4T^{2} \)
41 \( 1 - 180.T + 6.89e4T^{2} \)
43 \( 1 - 529. iT - 7.95e4T^{2} \)
47 \( 1 + 550. iT - 1.03e5T^{2} \)
53 \( 1 + 455. iT - 1.48e5T^{2} \)
59 \( 1 - 522.T + 2.05e5T^{2} \)
61 \( 1 + 536.T + 2.26e5T^{2} \)
67 \( 1 - 53.0iT - 3.00e5T^{2} \)
71 \( 1 - 401.T + 3.57e5T^{2} \)
73 \( 1 + 554. iT - 3.89e5T^{2} \)
79 \( 1 - 392.T + 4.93e5T^{2} \)
83 \( 1 + 1.41e3iT - 5.71e5T^{2} \)
89 \( 1 + 174.T + 7.04e5T^{2} \)
97 \( 1 + 223. iT - 9.12e5T^{2} \)
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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

−9.565815231969504233761102068196, −8.825174811830287522435469541945, −7.893394139699591002018913606610, −6.97855862437656851289711427652, −6.40301895644745299284350501710, −5.59653607990542512081474766800, −4.52722886103672398915648562690, −3.41514356303510281275168648543, −1.82791422947266331953848256960, −0.902303078572028902182789986287, 0.941984188196568303027669572236, 2.28406100632044054701023493780, 3.04329070537439741366896521377, 4.25780968057679145974519702539, 5.29811660314782657050676843297, 5.90978488730089275136505401026, 7.02836066579921662361516561652, 8.389664604015431527187937835009, 9.257861717205674447279844918651, 9.516954393586900297724526228965

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