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Properties

Label 2-930-155.92-c1-0-0
Degree $2$
Conductor $930$
Sign $0.872 - 0.488i$
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 + (−2.23 + 0.0168i)5-s + 1.00i·6-s + (−2.83 − 2.83i)7-s + (0.707 − 0.707i)8-s + 1.00i·9-s + (1.59 + 1.56i)10-s + 1.81i·11-s + (0.707 − 0.707i)12-s + (−3.73 − 3.73i)13-s + 4.01i·14-s + (1.59 + 1.56i)15-s − 1.00·16-s + (0.414 − 0.414i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.408 − 0.408i)3-s + 0.500i·4-s + (−0.999 + 0.00754i)5-s + 0.408i·6-s + (−1.07 − 1.07i)7-s + (0.250 − 0.250i)8-s + 0.333i·9-s + (0.503 + 0.496i)10-s + 0.546i·11-s + (0.204 − 0.204i)12-s + (−1.03 − 1.03i)13-s + 1.07i·14-s + (0.411 + 0.405i)15-s − 0.250·16-s + (0.100 − 0.100i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.872 - 0.488i$
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: \((2,\ 930,\ (\ :1/2),\ 0.872 - 0.488i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.294961 + 0.0769470i\)
\(L(\frac12)\) \(\approx\) \(0.294961 + 0.0769470i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 - 0.0168i)T \)
31 \( 1 + (-0.282 + 5.56i)T \)
good7 \( 1 + (2.83 + 2.83i)T + 7iT^{2} \)
11 \( 1 - 1.81iT - 11T^{2} \)
13 \( 1 + (3.73 + 3.73i)T + 13iT^{2} \)
17 \( 1 + (-0.414 + 0.414i)T - 17iT^{2} \)
19 \( 1 - 5.16iT - 19T^{2} \)
23 \( 1 + (1.45 + 1.45i)T + 23iT^{2} \)
29 \( 1 - 1.79T + 29T^{2} \)
37 \( 1 + (7.51 - 7.51i)T - 37iT^{2} \)
41 \( 1 - 8.55T + 41T^{2} \)
43 \( 1 + (1.19 + 1.19i)T + 43iT^{2} \)
47 \( 1 + (-5.64 - 5.64i)T + 47iT^{2} \)
53 \( 1 + (-10.0 - 10.0i)T + 53iT^{2} \)
59 \( 1 - 11.6iT - 59T^{2} \)
61 \( 1 + 14.9iT - 61T^{2} \)
67 \( 1 + (-8.55 - 8.55i)T + 67iT^{2} \)
71 \( 1 + 8.79T + 71T^{2} \)
73 \( 1 + (8.90 + 8.90i)T + 73iT^{2} \)
79 \( 1 - 1.10T + 79T^{2} \)
83 \( 1 + (-4.92 - 4.92i)T + 83iT^{2} \)
89 \( 1 + 2.43T + 89T^{2} \)
97 \( 1 + (-2.65 - 2.65i)T + 97iT^{2} \)
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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

−10.29318862566202623641480963149, −9.578170431185757186901747521172, −8.289195661492854041588039471173, −7.49650994500112235169060941740, −7.14268495916976016661933020204, −5.97613699748177152627687533595, −4.57740757157603432388029467730, −3.71242220433105427935181304106, −2.68261962364726726606294228717, −0.874102791881834745416196949765, 0.24612317414337087332373279673, 2.52898939998822675068008129959, 3.72066806733582847334937111425, 4.86770602965703367716536836718, 5.71538167711627698872509792124, 6.76018669552444007376458068231, 7.25348107738639591778354099350, 8.631148125698949311347443073108, 9.010013580507387881584103628973, 9.824338879929555175412971450946

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