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Properties

Label 2-930-15.2-c1-0-19
Degree $2$
Conductor $930$
Sign $-0.568 - 0.822i$
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 + (1.70 + 0.292i)3-s − 1.00i·4-s + (−1.19 + 1.88i)5-s + (−1.41 + 0.999i)6-s + (3.59 + 3.59i)7-s + (0.707 + 0.707i)8-s + (2.82 + i)9-s + (−0.489 − 2.18i)10-s + 1.67i·11-s + (0.292 − 1.70i)12-s + (−0.721 + 0.721i)13-s − 5.08·14-s + (−2.59 + 2.87i)15-s − 1.00·16-s + (−5.06 + 5.06i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.985 + 0.169i)3-s − 0.500i·4-s + (−0.535 + 0.844i)5-s + (−0.577 + 0.408i)6-s + (1.35 + 1.35i)7-s + (0.250 + 0.250i)8-s + (0.942 + 0.333i)9-s + (−0.154 − 0.689i)10-s + 0.503i·11-s + (0.0845 − 0.492i)12-s + (−0.200 + 0.200i)13-s − 1.35·14-s + (−0.670 + 0.742i)15-s − 0.250·16-s + (−1.22 + 1.22i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.568 - 0.822i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.568 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.568 - 0.822i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (497, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.568 - 0.822i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.797040 + 1.51903i\)
\(L(\frac12)\) \(\approx\) \(0.797040 + 1.51903i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.70 - 0.292i)T \)
5 \( 1 + (1.19 - 1.88i)T \)
31 \( 1 - T \)
good7 \( 1 + (-3.59 - 3.59i)T + 7iT^{2} \)
11 \( 1 - 1.67iT - 11T^{2} \)
13 \( 1 + (0.721 - 0.721i)T - 13iT^{2} \)
17 \( 1 + (5.06 - 5.06i)T - 17iT^{2} \)
19 \( 1 + 7.32iT - 19T^{2} \)
23 \( 1 + (3.01 + 3.01i)T + 23iT^{2} \)
29 \( 1 - 10.0T + 29T^{2} \)
37 \( 1 + (2 + 2i)T + 37iT^{2} \)
41 \( 1 + 10.6iT - 41T^{2} \)
43 \( 1 + (5.28 - 5.28i)T - 43iT^{2} \)
47 \( 1 + (1.86 - 1.86i)T - 47iT^{2} \)
53 \( 1 + (0.190 + 0.190i)T + 53iT^{2} \)
59 \( 1 + 3.00T + 59T^{2} \)
61 \( 1 - 9.83T + 61T^{2} \)
67 \( 1 + (-1.37 - 1.37i)T + 67iT^{2} \)
71 \( 1 - 1.32iT - 71T^{2} \)
73 \( 1 + (0.424 - 0.424i)T - 73iT^{2} \)
79 \( 1 + 1.39iT - 79T^{2} \)
83 \( 1 + (-7.37 - 7.37i)T + 83iT^{2} \)
89 \( 1 + 3.30T + 89T^{2} \)
97 \( 1 + (-5.96 - 5.96i)T + 97iT^{2} \)
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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

−10.31341363339413213768457479814, −9.189134920491220349998163543851, −8.486036017773256820725686259759, −8.165389349876092096645675892153, −7.09443379133647111612038414957, −6.39678462903431700823025372141, −4.93099799025551040915876637995, −4.27968718251276563259109607665, −2.60991063663461614211216008467, −2.02173657495387572995264952649, 0.894098459272542936382938132501, 1.84143773934817705429421597303, 3.36262334299914255444668725623, 4.26353740701282673210352689146, 4.90700020205313399769003004857, 6.79065331010101173450632188010, 7.73821508240958519282183828190, 8.138767542402484145136976931146, 8.709118372905724373850322026452, 9.821914205396765445832205253429

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