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Properties

Label 2-1530-15.2-c1-0-17
Degree $2$
Conductor $1530$
Sign $-0.0772 + 0.997i$
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 + (−2.23 − 0.0345i)5-s + (0.796 + 0.796i)7-s + (−0.707 − 0.707i)8-s + (−1.60 + 1.55i)10-s + 2.14i·11-s + (3.95 − 3.95i)13-s + 1.12·14-s − 1.00·16-s + (0.707 − 0.707i)17-s − 4.73i·19-s + (−0.0345 + 2.23i)20-s + (1.51 + 1.51i)22-s + (5.76 + 5.76i)23-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s − 0.500i·4-s + (−0.999 − 0.0154i)5-s + (0.301 + 0.301i)7-s + (−0.250 − 0.250i)8-s + (−0.507 + 0.492i)10-s + 0.647i·11-s + (1.09 − 1.09i)13-s + 0.301·14-s − 0.250·16-s + (0.171 − 0.171i)17-s − 1.08i·19-s + (−0.00771 + 0.499i)20-s + (0.323 + 0.323i)22-s + (1.20 + 1.20i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0772 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0772 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1530\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 17\)
Sign: $-0.0772 + 0.997i$
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: \((2,\ 1530,\ (\ :1/2),\ -0.0772 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.804458231\)
\(L(\frac12)\) \(\approx\) \(1.804458231\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 + 0.0345i)T \)
17 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 + (-0.796 - 0.796i)T + 7iT^{2} \)
11 \( 1 - 2.14iT - 11T^{2} \)
13 \( 1 + (-3.95 + 3.95i)T - 13iT^{2} \)
19 \( 1 + 4.73iT - 19T^{2} \)
23 \( 1 + (-5.76 - 5.76i)T + 23iT^{2} \)
29 \( 1 + 6.62T + 29T^{2} \)
31 \( 1 + 1.21T + 31T^{2} \)
37 \( 1 + (8.37 + 8.37i)T + 37iT^{2} \)
41 \( 1 + 9.89iT - 41T^{2} \)
43 \( 1 + (-2.47 + 2.47i)T - 43iT^{2} \)
47 \( 1 + (-8.42 + 8.42i)T - 47iT^{2} \)
53 \( 1 + (-0.461 - 0.461i)T + 53iT^{2} \)
59 \( 1 - 4.07T + 59T^{2} \)
61 \( 1 - 11.7T + 61T^{2} \)
67 \( 1 + (-2.22 - 2.22i)T + 67iT^{2} \)
71 \( 1 - 3.51iT - 71T^{2} \)
73 \( 1 + (3.15 - 3.15i)T - 73iT^{2} \)
79 \( 1 + 16.2iT - 79T^{2} \)
83 \( 1 + (11.0 + 11.0i)T + 83iT^{2} \)
89 \( 1 - 10.8T + 89T^{2} \)
97 \( 1 + (-1.47 - 1.47i)T + 97iT^{2} \)
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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

−9.051782000519627928832760084798, −8.695166274431123845453600253854, −7.41424648770899514147739169728, −7.08501821560869261994465684250, −5.53417336875945551842516149779, −5.19529095078337794494148788650, −3.90038807019543255011484630244, −3.41009378048333910759851474219, −2.14485701332222457583869324719, −0.68738575511562161013649652232, 1.28318820244672910803970190980, 3.03663336176597782819724987197, 3.90012738541866136367224678741, 4.48849499000535060972980416991, 5.59007585605357846086726956909, 6.51649350092834302700138355830, 7.14128737054265911146434861357, 8.191294118960905628901337095190, 8.474655772130485506886242745143, 9.464591253688783456759086011129

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