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Properties

Label 2-930-5.4-c3-0-82
Degree $2$
Conductor $930$
Sign $-0.925 + 0.379i$
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 + (−10.3 + 4.24i)5-s + 6·6-s − 36.3i·7-s − 8i·8-s − 9·9-s + (−8.49 − 20.6i)10-s + 18.7·11-s + 12i·12-s − 53.3i·13-s + 72.7·14-s + (12.7 + 31.0i)15-s + 16·16-s − 42.5i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (−0.925 + 0.379i)5-s + 0.408·6-s − 1.96i·7-s − 0.353i·8-s − 0.333·9-s + (−0.268 − 0.654i)10-s + 0.514·11-s + 0.288i·12-s − 1.13i·13-s + 1.38·14-s + (0.219 + 0.534i)15-s + 0.250·16-s − 0.607i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 + 0.379i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.925 + 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.925 + 0.379i$
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: \((2,\ 930,\ (\ :3/2),\ -0.925 + 0.379i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9049707308\)
\(L(\frac12)\) \(\approx\) \(0.9049707308\)
\(L(\frac{5}{2})\) 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 \( 1 + (10.3 - 4.24i)T \)
31 \( 1 + 31T \)
good7 \( 1 + 36.3iT - 343T^{2} \)
11 \( 1 - 18.7T + 1.33e3T^{2} \)
13 \( 1 + 53.3iT - 2.19e3T^{2} \)
17 \( 1 + 42.5iT - 4.91e3T^{2} \)
19 \( 1 - 45.0T + 6.85e3T^{2} \)
23 \( 1 + 27.3iT - 1.21e4T^{2} \)
29 \( 1 - 20.0T + 2.43e4T^{2} \)
37 \( 1 + 126. iT - 5.06e4T^{2} \)
41 \( 1 - 208.T + 6.89e4T^{2} \)
43 \( 1 + 171. iT - 7.95e4T^{2} \)
47 \( 1 - 108. iT - 1.03e5T^{2} \)
53 \( 1 - 193. iT - 1.48e5T^{2} \)
59 \( 1 - 147.T + 2.05e5T^{2} \)
61 \( 1 + 644.T + 2.26e5T^{2} \)
67 \( 1 - 34.1iT - 3.00e5T^{2} \)
71 \( 1 + 390.T + 3.57e5T^{2} \)
73 \( 1 + 593. iT - 3.89e5T^{2} \)
79 \( 1 - 134.T + 4.93e5T^{2} \)
83 \( 1 + 652. iT - 5.71e5T^{2} \)
89 \( 1 - 551.T + 7.04e5T^{2} \)
97 \( 1 + 805. iT - 9.12e5T^{2} \)
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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

−9.157114935807258214417444739240, −7.995207712866734082212016213523, −7.51081023280563062044590342000, −7.06017454216264941286204411323, −6.14433315226546151709004844463, −4.83022021611922912742377733767, −3.93839048437549233215199666027, −3.11584882875105300729130158464, −1.01220381531110687661817983363, −0.28895427556708047612013259443, 1.54100264902557631444581486320, 2.72321598170774458645711453737, 3.72342069910290247176370969888, 4.62442031567449253709005104116, 5.45858871038143125170282886446, 6.45050391156809727424552665280, 7.88287295248121949528217265504, 8.783632192366910627812906923712, 9.095915861778594670896119603750, 9.890014541931356656935869492782

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