Properties

Label 2-930-5.4-c3-0-63
Degree $2$
Conductor $930$
Sign $0.147 + 0.988i$
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 + (1.65 + 11.0i)5-s + 6·6-s − 2.45i·7-s + 8i·8-s − 9·9-s + (22.1 − 3.30i)10-s − 35.2·11-s − 12i·12-s − 31.1i·13-s − 4.90·14-s + (−33.1 + 4.96i)15-s + 16·16-s − 47.1i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (0.147 + 0.988i)5-s + 0.408·6-s − 0.132i·7-s + 0.353i·8-s − 0.333·9-s + (0.699 − 0.104i)10-s − 0.966·11-s − 0.288i·12-s − 0.664i·13-s − 0.0936·14-s + (−0.570 + 0.0854i)15-s + 0.250·16-s − 0.672i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.147 + 0.988i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.147 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.147 + 0.988i$
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: \((2,\ 930,\ (\ :3/2),\ 0.147 + 0.988i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.197539569\)
\(L(\frac12)\) \(\approx\) \(1.197539569\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.65 - 11.0i)T \)
31 \( 1 + 31T \)
good7 \( 1 + 2.45iT - 343T^{2} \)
11 \( 1 + 35.2T + 1.33e3T^{2} \)
13 \( 1 + 31.1iT - 2.19e3T^{2} \)
17 \( 1 + 47.1iT - 4.91e3T^{2} \)
19 \( 1 - 118.T + 6.85e3T^{2} \)
23 \( 1 - 57.1iT - 1.21e4T^{2} \)
29 \( 1 + 298.T + 2.43e4T^{2} \)
37 \( 1 - 82.3iT - 5.06e4T^{2} \)
41 \( 1 + 25.0T + 6.89e4T^{2} \)
43 \( 1 + 264. iT - 7.95e4T^{2} \)
47 \( 1 + 344. iT - 1.03e5T^{2} \)
53 \( 1 + 422. iT - 1.48e5T^{2} \)
59 \( 1 + 43.3T + 2.05e5T^{2} \)
61 \( 1 - 96.2T + 2.26e5T^{2} \)
67 \( 1 - 706. iT - 3.00e5T^{2} \)
71 \( 1 - 1.11e3T + 3.57e5T^{2} \)
73 \( 1 - 94.3iT - 3.89e5T^{2} \)
79 \( 1 - 607.T + 4.93e5T^{2} \)
83 \( 1 + 604. iT - 5.71e5T^{2} \)
89 \( 1 - 391.T + 7.04e5T^{2} \)
97 \( 1 + 1.04e3iT - 9.12e5T^{2} \)
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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

−9.945768324015367030323493290964, −8.920634101570805012097639893572, −7.74609791848292447472063050678, −7.16333686118814112300886427521, −5.63274845321048284429915644227, −5.19939563366577887895553167890, −3.71898070272629752740424142612, −3.12277667922324877400112317793, −2.11265631625209918172105208362, −0.36273976594442804388704476972, 0.962557693420458611056782686070, 2.17797318069810423981112027468, 3.71068418401433464463208897498, 4.89653002416277186038849270061, 5.56914559482689999236412025907, 6.37592096756525257943022488819, 7.61779419424044471664978855349, 7.893502746730032005153012364562, 9.048878996022357080868466525036, 9.441029049130530920099225308513

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