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Properties

Label 2-930-5.4-c3-0-6
Degree $2$
Conductor $930$
Sign $0.196 - 0.980i$
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 + (2.19 − 10.9i)5-s + 6·6-s − 7.05i·7-s + 8i·8-s − 9·9-s + (−21.9 − 4.39i)10-s − 11.6·11-s − 12i·12-s − 71.0i·13-s − 14.1·14-s + (32.8 + 6.59i)15-s + 16·16-s + 133. i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (0.196 − 0.980i)5-s + 0.408·6-s − 0.381i·7-s + 0.353i·8-s − 0.333·9-s + (−0.693 − 0.139i)10-s − 0.319·11-s − 0.288i·12-s − 1.51i·13-s − 0.269·14-s + (0.566 + 0.113i)15-s + 0.250·16-s + 1.90i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.196 - 0.980i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.196 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.196 - 0.980i$
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: \((2,\ 930,\ (\ :3/2),\ 0.196 - 0.980i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4574884035\)
\(L(\frac12)\) \(\approx\) \(0.4574884035\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.19 + 10.9i)T \)
31 \( 1 + 31T \)
good7 \( 1 + 7.05iT - 343T^{2} \)
11 \( 1 + 11.6T + 1.33e3T^{2} \)
13 \( 1 + 71.0iT - 2.19e3T^{2} \)
17 \( 1 - 133. iT - 4.91e3T^{2} \)
19 \( 1 + 101.T + 6.85e3T^{2} \)
23 \( 1 - 38.2iT - 1.21e4T^{2} \)
29 \( 1 + 24.1T + 2.43e4T^{2} \)
37 \( 1 + 270. iT - 5.06e4T^{2} \)
41 \( 1 - 64.4T + 6.89e4T^{2} \)
43 \( 1 - 298. iT - 7.95e4T^{2} \)
47 \( 1 + 166. iT - 1.03e5T^{2} \)
53 \( 1 - 386. iT - 1.48e5T^{2} \)
59 \( 1 - 796.T + 2.05e5T^{2} \)
61 \( 1 + 671.T + 2.26e5T^{2} \)
67 \( 1 - 525. iT - 3.00e5T^{2} \)
71 \( 1 - 19.0T + 3.57e5T^{2} \)
73 \( 1 - 723. iT - 3.89e5T^{2} \)
79 \( 1 + 800.T + 4.93e5T^{2} \)
83 \( 1 + 826. iT - 5.71e5T^{2} \)
89 \( 1 + 166.T + 7.04e5T^{2} \)
97 \( 1 - 1.59e3iT - 9.12e5T^{2} \)
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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

−10.22299892711393970761328023386, −9.045305278759745287817798184166, −8.420239383181270670960699742039, −7.71803913286005756109221631096, −6.01804685043070057523857179699, −5.42125828813030005698431020001, −4.31998937569051523769365786178, −3.70603550313925906958501011566, −2.35226737751539574531380119626, −1.09877254237335493837847113757, 0.12456444729602490117011889682, 1.96921730631612939727832628055, 2.88164328349395597536297221284, 4.26528104265036466417576941726, 5.31946112815764801453100039628, 6.37449657470394363780852398725, 6.86744605031395155748668992230, 7.53677786795499435860624837739, 8.624086641333903181663868825173, 9.317975389247206230195908250932

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