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Properties

Label 2-930-5.4-c3-0-4
Degree $2$
Conductor $930$
Sign $-0.664 + 0.747i$
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 + (−7.42 + 8.35i)5-s + 6·6-s + 7.62i·7-s + 8i·8-s − 9·9-s + (16.7 + 14.8i)10-s − 39.0·11-s − 12i·12-s + 55.9i·13-s + 15.2·14-s + (−25.0 − 22.2i)15-s + 16·16-s + 80.2i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (−0.664 + 0.747i)5-s + 0.408·6-s + 0.411i·7-s + 0.353i·8-s − 0.333·9-s + (0.528 + 0.469i)10-s − 1.06·11-s − 0.288i·12-s + 1.19i·13-s + 0.291·14-s + (−0.431 − 0.383i)15-s + 0.250·16-s + 1.14i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.664 + 0.747i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.664 + 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.664 + 0.747i$
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: \((2,\ 930,\ (\ :3/2),\ -0.664 + 0.747i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1877218746\)
\(L(\frac12)\) \(\approx\) \(0.1877218746\)
\(L(\frac{5}{2})\) 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 \( 1 + (7.42 - 8.35i)T \)
31 \( 1 + 31T \)
good7 \( 1 - 7.62iT - 343T^{2} \)
11 \( 1 + 39.0T + 1.33e3T^{2} \)
13 \( 1 - 55.9iT - 2.19e3T^{2} \)
17 \( 1 - 80.2iT - 4.91e3T^{2} \)
19 \( 1 + 72.3T + 6.85e3T^{2} \)
23 \( 1 - 51.5iT - 1.21e4T^{2} \)
29 \( 1 - 59.4T + 2.43e4T^{2} \)
37 \( 1 + 12.6iT - 5.06e4T^{2} \)
41 \( 1 + 255.T + 6.89e4T^{2} \)
43 \( 1 - 62.3iT - 7.95e4T^{2} \)
47 \( 1 - 625. iT - 1.03e5T^{2} \)
53 \( 1 + 253. iT - 1.48e5T^{2} \)
59 \( 1 - 286.T + 2.05e5T^{2} \)
61 \( 1 - 297.T + 2.26e5T^{2} \)
67 \( 1 + 342. iT - 3.00e5T^{2} \)
71 \( 1 - 399.T + 3.57e5T^{2} \)
73 \( 1 + 612. iT - 3.89e5T^{2} \)
79 \( 1 + 623.T + 4.93e5T^{2} \)
83 \( 1 + 440. iT - 5.71e5T^{2} \)
89 \( 1 + 409.T + 7.04e5T^{2} \)
97 \( 1 + 819. iT - 9.12e5T^{2} \)
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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

−10.42344803151108856198749042764, −9.556056716196212285571916612592, −8.592965755124207789572973951212, −7.963470359925673339163764287335, −6.78368837060320718469706150608, −5.80139648084342295492761045540, −4.62839636272491974172654874051, −3.89481717194440154221994439183, −2.91020376478904564806099040251, −1.93468122631330515742633428229, 0.06420961086564212873280481911, 0.817083235342279949668562308920, 2.59089722846465147503059140785, 3.84552692065898479758309884632, 5.00358414477942514049432211757, 5.51517526513273479995292873205, 6.82699063117106560457065176689, 7.47640032911909948494567259475, 8.247970331089338396832498226231, 8.693524106351262593718275105474

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