Properties

Label 2-930-155.42-c1-0-30
Degree $2$
Conductor $930$
Sign $-0.935 + 0.353i$
Analytic cond. $7.42608$
Root an. cond. $2.72508$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.453 − 0.891i)2-s + (−0.0523 + 0.998i)3-s + (−0.587 − 0.809i)4-s + (−0.971 − 2.01i)5-s + (0.866 + 0.5i)6-s + (2.64 − 3.26i)7-s + (−0.987 + 0.156i)8-s + (−0.994 − 0.104i)9-s + (−2.23 − 0.0483i)10-s + (0.903 − 2.03i)11-s + (0.838 − 0.544i)12-s + (−5.44 − 3.53i)13-s + (−1.70 − 3.84i)14-s + (2.06 − 0.865i)15-s + (−0.309 + 0.951i)16-s + (−1.47 + 3.84i)17-s + ⋯
L(s)  = 1  + (0.321 − 0.630i)2-s + (−0.0302 + 0.576i)3-s + (−0.293 − 0.404i)4-s + (−0.434 − 0.900i)5-s + (0.353 + 0.204i)6-s + (0.999 − 1.23i)7-s + (−0.349 + 0.0553i)8-s + (−0.331 − 0.0348i)9-s + (−0.706 − 0.0153i)10-s + (0.272 − 0.612i)11-s + (0.242 − 0.157i)12-s + (−1.50 − 0.980i)13-s + (−0.456 − 1.02i)14-s + (0.532 − 0.223i)15-s + (−0.0772 + 0.237i)16-s + (−0.358 + 0.933i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 + 0.353i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.935 + 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.935 + 0.353i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (817, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.935 + 0.353i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.222548 - 1.21866i\)
\(L(\frac12)\) \(\approx\) \(0.222548 - 1.21866i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.453 + 0.891i)T \)
3 \( 1 + (0.0523 - 0.998i)T \)
5 \( 1 + (0.971 + 2.01i)T \)
31 \( 1 + (-1.12 + 5.45i)T \)
good7 \( 1 + (-2.64 + 3.26i)T + (-1.45 - 6.84i)T^{2} \)
11 \( 1 + (-0.903 + 2.03i)T + (-7.36 - 8.17i)T^{2} \)
13 \( 1 + (5.44 + 3.53i)T + (5.28 + 11.8i)T^{2} \)
17 \( 1 + (1.47 - 3.84i)T + (-12.6 - 11.3i)T^{2} \)
19 \( 1 + (0.913 - 4.29i)T + (-17.3 - 7.72i)T^{2} \)
23 \( 1 + (-1.25 - 7.94i)T + (-21.8 + 7.10i)T^{2} \)
29 \( 1 + (2.43 + 7.49i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 + (2.59 + 9.69i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (2.42 - 2.69i)T + (-4.28 - 40.7i)T^{2} \)
43 \( 1 + (3.53 + 5.44i)T + (-17.4 + 39.2i)T^{2} \)
47 \( 1 + (-9.67 + 4.93i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (3.78 - 3.06i)T + (11.0 - 51.8i)T^{2} \)
59 \( 1 + (-2.46 + 2.21i)T + (6.16 - 58.6i)T^{2} \)
61 \( 1 + 5.21iT - 61T^{2} \)
67 \( 1 + (1.69 - 6.31i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (0.524 - 4.98i)T + (-69.4 - 14.7i)T^{2} \)
73 \( 1 + (-0.537 - 1.39i)T + (-54.2 + 48.8i)T^{2} \)
79 \( 1 + (-3.91 + 1.74i)T + (52.8 - 58.7i)T^{2} \)
83 \( 1 + (-6.15 + 0.322i)T + (82.5 - 8.67i)T^{2} \)
89 \( 1 + (-11.9 + 8.70i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-2.57 - 0.407i)T + (92.2 + 29.9i)T^{2} \)
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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

−9.940262363708510440007326767134, −8.996551564096758717549517960230, −7.952551107117784225560014268421, −7.54678235501826435706781479747, −5.76136314562410642135313006149, −5.10721091815354887034131040948, −4.10751101793180638144065095896, −3.69751546371348737099429067735, −1.90684588328775685370873605386, −0.51126778665056351456975941900, 2.15475041765430183216631473627, 2.87250946609989845369198517296, 4.73151361388536998455664542598, 4.94342857137827770653234115660, 6.52812775446016511057999462420, 6.91232799415536969758150745303, 7.67137025656938621358777157106, 8.685189881131019782321550812332, 9.252726835502483840804071844287, 10.56850622100373159328403119852

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