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Properties

Label 2-930-155.92-c1-0-30
Degree $2$
Conductor $930$
Sign $-0.895 + 0.444i$
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 + (1.06 − 1.96i)5-s − 1.00i·6-s + (−1.05 − 1.05i)7-s + (0.707 − 0.707i)8-s + 1.00i·9-s + (−2.14 + 0.633i)10-s − 5.04i·11-s + (−0.707 + 0.707i)12-s + (−4.63 − 4.63i)13-s + 1.49i·14-s + (2.14 − 0.633i)15-s − 1.00·16-s + (−2.53 + 2.53i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (0.408 + 0.408i)3-s + 0.500i·4-s + (0.477 − 0.878i)5-s − 0.408i·6-s + (−0.400 − 0.400i)7-s + (0.250 − 0.250i)8-s + 0.333i·9-s + (−0.678 + 0.200i)10-s − 1.52i·11-s + (−0.204 + 0.204i)12-s + (−1.28 − 1.28i)13-s + 0.400i·14-s + (0.553 − 0.163i)15-s − 0.250·16-s + (−0.613 + 0.613i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.895 + 0.444i$
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: \((2,\ 930,\ (\ :1/2),\ -0.895 + 0.444i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.190614 - 0.813735i\)
\(L(\frac12)\) \(\approx\) \(0.190614 - 0.813735i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.06 + 1.96i)T \)
31 \( 1 + (5.11 - 2.19i)T \)
good7 \( 1 + (1.05 + 1.05i)T + 7iT^{2} \)
11 \( 1 + 5.04iT - 11T^{2} \)
13 \( 1 + (4.63 + 4.63i)T + 13iT^{2} \)
17 \( 1 + (2.53 - 2.53i)T - 17iT^{2} \)
19 \( 1 - 7.83iT - 19T^{2} \)
23 \( 1 + (2.05 + 2.05i)T + 23iT^{2} \)
29 \( 1 + 3.01T + 29T^{2} \)
37 \( 1 + (-2.29 + 2.29i)T - 37iT^{2} \)
41 \( 1 - 2.23T + 41T^{2} \)
43 \( 1 + (4.87 + 4.87i)T + 43iT^{2} \)
47 \( 1 + (-5.72 - 5.72i)T + 47iT^{2} \)
53 \( 1 + (-5.97 - 5.97i)T + 53iT^{2} \)
59 \( 1 + 5.06iT - 59T^{2} \)
61 \( 1 + 4.63iT - 61T^{2} \)
67 \( 1 + (5.44 + 5.44i)T + 67iT^{2} \)
71 \( 1 - 15.4T + 71T^{2} \)
73 \( 1 + (5.66 + 5.66i)T + 73iT^{2} \)
79 \( 1 - 12.7T + 79T^{2} \)
83 \( 1 + (8.45 + 8.45i)T + 83iT^{2} \)
89 \( 1 + 2.92T + 89T^{2} \)
97 \( 1 + (7.21 + 7.21i)T + 97iT^{2} \)
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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

−9.812762056454512261326466482131, −8.914370775514891555729648125511, −8.257775573614162540303448054681, −7.59265444086585366916099757214, −6.07461207028842914020467942771, −5.34719879778714520574489456915, −4.05591886681216097309986974390, −3.23400423241300693475047503733, −1.97378534797357981239636683785, −0.40552438934602185566252235242, 2.08078111949364329188914234682, 2.55142347164984218394314868713, 4.31126276894695852558196899326, 5.33451572976038858883276241171, 6.66761884262515743599539756542, 7.00281247967262097441683675200, 7.55634396857171187144180553114, 9.071410341019221864559384124565, 9.437182256553322114216632024527, 10.00062326565332767315068997168

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