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Properties

Label 2-930-155.92-c1-0-18
Degree $2$
Conductor $930$
Sign $-0.686 - 0.727i$
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  + (0.707 + 0.707i)2-s + (0.707 + 0.707i)3-s + 1.00i·4-s + (0.0490 + 2.23i)5-s + 1.00i·6-s + (3.32 + 3.32i)7-s + (−0.707 + 0.707i)8-s + 1.00i·9-s + (−1.54 + 1.61i)10-s + 4.17i·11-s + (−0.707 + 0.707i)12-s + (−2.60 − 2.60i)13-s + 4.70i·14-s + (−1.54 + 1.61i)15-s − 1.00·16-s + (4.75 − 4.75i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.408 + 0.408i)3-s + 0.500i·4-s + (0.0219 + 0.999i)5-s + 0.408i·6-s + (1.25 + 1.25i)7-s + (−0.250 + 0.250i)8-s + 0.333i·9-s + (−0.488 + 0.510i)10-s + 1.25i·11-s + (−0.204 + 0.204i)12-s + (−0.722 − 0.722i)13-s + 1.25i·14-s + (−0.399 + 0.417i)15-s − 0.250·16-s + (1.15 − 1.15i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.686 - 0.727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.686 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.686 - 0.727i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (247, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.686 - 0.727i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03168 + 2.39143i\)
\(L(\frac12)\) \(\approx\) \(1.03168 + 2.39143i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (-0.0490 - 2.23i)T \)
31 \( 1 + (1.00 + 5.47i)T \)
good7 \( 1 + (-3.32 - 3.32i)T + 7iT^{2} \)
11 \( 1 - 4.17iT - 11T^{2} \)
13 \( 1 + (2.60 + 2.60i)T + 13iT^{2} \)
17 \( 1 + (-4.75 + 4.75i)T - 17iT^{2} \)
19 \( 1 + 7.18iT - 19T^{2} \)
23 \( 1 + (4.10 + 4.10i)T + 23iT^{2} \)
29 \( 1 - 6.16T + 29T^{2} \)
37 \( 1 + (3.18 - 3.18i)T - 37iT^{2} \)
41 \( 1 - 8.63T + 41T^{2} \)
43 \( 1 + (-0.501 - 0.501i)T + 43iT^{2} \)
47 \( 1 + (-0.354 - 0.354i)T + 47iT^{2} \)
53 \( 1 + (-5.87 - 5.87i)T + 53iT^{2} \)
59 \( 1 - 2.54iT - 59T^{2} \)
61 \( 1 + 5.63iT - 61T^{2} \)
67 \( 1 + (-7.48 - 7.48i)T + 67iT^{2} \)
71 \( 1 + 13.4T + 71T^{2} \)
73 \( 1 + (3.37 + 3.37i)T + 73iT^{2} \)
79 \( 1 + 9.67T + 79T^{2} \)
83 \( 1 + (-6.43 - 6.43i)T + 83iT^{2} \)
89 \( 1 + 3.82T + 89T^{2} \)
97 \( 1 + (4.24 + 4.24i)T + 97iT^{2} \)
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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

−10.25851999025292066388201961895, −9.549682568679438989618822904433, −8.574797784612879497086077679392, −7.65513298395185107520680038936, −7.20249260042445624136949884106, −5.91395425200017818802254256673, −5.02570314912398286653118210858, −4.43569686144115983165302139007, −2.74611514302912504060358377721, −2.43822130497354683695809807877, 1.11407187631147328940290541141, 1.77395834958850873026502041975, 3.60952558554813902986027604300, 4.15803852414740462426704957990, 5.27448630016547314245247180532, 6.06045056455856913496825504664, 7.49292058390929652856965763304, 8.079583395038236303216131556207, 8.753649314250923751914838134904, 10.00381996109247843669883297624

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