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Properties

Label 2-1530-15.2-c1-0-16
Degree $2$
Conductor $1530$
Sign $0.197 + 0.980i$
Analytic cond. $12.2171$
Root an. cond. $3.49529$
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 − 1.00i·4-s + (−2.16 + 0.577i)5-s + (1.51 + 1.51i)7-s + (0.707 + 0.707i)8-s + (1.11 − 1.93i)10-s − 5.47i·11-s + (1.53 − 1.53i)13-s − 2.14·14-s − 1.00·16-s + (−0.707 + 0.707i)17-s + 8.66i·19-s + (0.577 + 2.16i)20-s + (3.87 + 3.87i)22-s + (−4.15 − 4.15i)23-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (−0.966 + 0.258i)5-s + (0.573 + 0.573i)7-s + (0.250 + 0.250i)8-s + (0.353 − 0.612i)10-s − 1.65i·11-s + (0.426 − 0.426i)13-s − 0.573·14-s − 0.250·16-s + (−0.171 + 0.171i)17-s + 1.98i·19-s + (0.129 + 0.483i)20-s + (0.825 + 0.825i)22-s + (−0.865 − 0.865i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.197 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.197 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1530\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 17\)
Sign: $0.197 + 0.980i$
Analytic conductor: \(12.2171\)
Root analytic conductor: \(3.49529\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1530} (647, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1530,\ (\ :1/2),\ 0.197 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5892036220\)
\(L(\frac12)\) \(\approx\) \(0.5892036220\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (2.16 - 0.577i)T \)
17 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 + (-1.51 - 1.51i)T + 7iT^{2} \)
11 \( 1 + 5.47iT - 11T^{2} \)
13 \( 1 + (-1.53 + 1.53i)T - 13iT^{2} \)
19 \( 1 - 8.66iT - 19T^{2} \)
23 \( 1 + (4.15 + 4.15i)T + 23iT^{2} \)
29 \( 1 + 3.13T + 29T^{2} \)
31 \( 1 + 9.82T + 31T^{2} \)
37 \( 1 + (4.21 + 4.21i)T + 37iT^{2} \)
41 \( 1 - 3.48iT - 41T^{2} \)
43 \( 1 + (-6.41 + 6.41i)T - 43iT^{2} \)
47 \( 1 + (-2.96 + 2.96i)T - 47iT^{2} \)
53 \( 1 + (3.58 + 3.58i)T + 53iT^{2} \)
59 \( 1 - 11.1T + 59T^{2} \)
61 \( 1 - 4.75T + 61T^{2} \)
67 \( 1 + (2.86 + 2.86i)T + 67iT^{2} \)
71 \( 1 - 1.20iT - 71T^{2} \)
73 \( 1 + (0.699 - 0.699i)T - 73iT^{2} \)
79 \( 1 + 12.2iT - 79T^{2} \)
83 \( 1 + (5.76 + 5.76i)T + 83iT^{2} \)
89 \( 1 + 9.65T + 89T^{2} \)
97 \( 1 + (9.41 + 9.41i)T + 97iT^{2} \)
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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

−8.852910600710271376783399689575, −8.416037513831983898568660163421, −7.928659237479382583071899665842, −7.02674659049454337832137640516, −5.79590283024239706955368326834, −5.63683619651884768638510926588, −4.06387538994240247823810806598, −3.38429829181151684442191670911, −1.87421100685270797346894532362, −0.29642755970231264091092041710, 1.27248313182490194328917585275, 2.39393542762161320555560494410, 3.81043342576875586624905721922, 4.38055592210796245430630385138, 5.20403145606213524552549882745, 6.91120205614862687028310303165, 7.31434568837214356428215726916, 7.980003984765676881723664900407, 9.044706731141496352234698038058, 9.441738423173833992738142976055

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