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Properties

Label 2-930-15.2-c1-0-36
Degree $2$
Conductor $930$
Sign $0.971 - 0.236i$
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 + (1.69 + 0.333i)3-s − 1.00i·4-s + (1.37 + 1.76i)5-s + (−1.43 + 0.965i)6-s + (−2.07 − 2.07i)7-s + (0.707 + 0.707i)8-s + (2.77 + 1.13i)9-s + (−2.21 − 0.270i)10-s − 3.16i·11-s + (0.333 − 1.69i)12-s + (4.31 − 4.31i)13-s + 2.93·14-s + (1.75 + 3.45i)15-s − 1.00·16-s + (1.60 − 1.60i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.981 + 0.192i)3-s − 0.500i·4-s + (0.616 + 0.787i)5-s + (−0.587 + 0.394i)6-s + (−0.783 − 0.783i)7-s + (0.250 + 0.250i)8-s + (0.925 + 0.378i)9-s + (−0.701 − 0.0855i)10-s − 0.955i·11-s + (0.0963 − 0.490i)12-s + (1.19 − 1.19i)13-s + 0.783·14-s + (0.453 + 0.891i)15-s − 0.250·16-s + (0.389 − 0.389i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.236i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 - 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.971 - 0.236i$
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: \((2,\ 930,\ (\ :1/2),\ 0.971 - 0.236i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.87684 + 0.224750i\)
\(L(\frac12)\) \(\approx\) \(1.87684 + 0.224750i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.69 - 0.333i)T \)
5 \( 1 + (-1.37 - 1.76i)T \)
31 \( 1 + T \)
good7 \( 1 + (2.07 + 2.07i)T + 7iT^{2} \)
11 \( 1 + 3.16iT - 11T^{2} \)
13 \( 1 + (-4.31 + 4.31i)T - 13iT^{2} \)
17 \( 1 + (-1.60 + 1.60i)T - 17iT^{2} \)
19 \( 1 + 4.20iT - 19T^{2} \)
23 \( 1 + (1.74 + 1.74i)T + 23iT^{2} \)
29 \( 1 - 6.90T + 29T^{2} \)
37 \( 1 + (-5.96 - 5.96i)T + 37iT^{2} \)
41 \( 1 - 9.79iT - 41T^{2} \)
43 \( 1 + (7.01 - 7.01i)T - 43iT^{2} \)
47 \( 1 + (3.86 - 3.86i)T - 47iT^{2} \)
53 \( 1 + (5.48 + 5.48i)T + 53iT^{2} \)
59 \( 1 + 1.65T + 59T^{2} \)
61 \( 1 + 5.62T + 61T^{2} \)
67 \( 1 + (11.1 + 11.1i)T + 67iT^{2} \)
71 \( 1 - 6.44iT - 71T^{2} \)
73 \( 1 + (-11.3 + 11.3i)T - 73iT^{2} \)
79 \( 1 - 11.1iT - 79T^{2} \)
83 \( 1 + (-5.80 - 5.80i)T + 83iT^{2} \)
89 \( 1 - 12.5T + 89T^{2} \)
97 \( 1 + (6.14 + 6.14i)T + 97iT^{2} \)
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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

−9.943462874675479907151230930155, −9.361486345274176990182795071073, −8.272242439404314866687587838198, −7.81560887150183278269125259014, −6.55794706332575240669727099083, −6.28092636945760322763721424740, −4.84094316792356100781668109089, −3.33547103126843840612697211198, −2.93152201865353101404771338513, −1.07787688523126281759380452339, 1.51463539830224757924103641870, 2.20416969542649143496885568557, 3.51952460510136824892804857675, 4.38609827139457848068421211229, 5.86410125818296003129107704582, 6.68815662025708744782488857222, 7.82567260706729145969564395295, 8.718215787085675999120998911864, 9.127145869531202676850306869047, 9.804956038520526879865395875236

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