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Properties

Label 2-1530-15.2-c1-0-23
Degree $2$
Conductor $1530$
Sign $-0.864 + 0.502i$
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 + (−1.24 − 1.85i)5-s + (−1.28 − 1.28i)7-s + (0.707 + 0.707i)8-s + (2.19 + 0.437i)10-s + 1.23i·11-s + (3.04 − 3.04i)13-s + 1.82·14-s − 1.00·16-s + (−0.707 + 0.707i)17-s − 3.83i·19-s + (−1.85 + 1.24i)20-s + (−0.874 − 0.874i)22-s + (−0.795 − 0.795i)23-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (−0.555 − 0.831i)5-s + (−0.486 − 0.486i)7-s + (0.250 + 0.250i)8-s + (0.693 + 0.138i)10-s + 0.372i·11-s + (0.843 − 0.843i)13-s + 0.486·14-s − 0.250·16-s + (−0.171 + 0.171i)17-s − 0.879i·19-s + (−0.415 + 0.277i)20-s + (−0.186 − 0.186i)22-s + (−0.165 − 0.165i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.864 + 0.502i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.864 + 0.502i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1530\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 17\)
Sign: $-0.864 + 0.502i$
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: \((2,\ 1530,\ (\ :1/2),\ -0.864 + 0.502i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4217236742\)
\(L(\frac12)\) \(\approx\) \(0.4217236742\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.24 + 1.85i)T \)
17 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 + (1.28 + 1.28i)T + 7iT^{2} \)
11 \( 1 - 1.23iT - 11T^{2} \)
13 \( 1 + (-3.04 + 3.04i)T - 13iT^{2} \)
19 \( 1 + 3.83iT - 19T^{2} \)
23 \( 1 + (0.795 + 0.795i)T + 23iT^{2} \)
29 \( 1 + 0.0783T + 29T^{2} \)
31 \( 1 - 3.53T + 31T^{2} \)
37 \( 1 + (2.05 + 2.05i)T + 37iT^{2} \)
41 \( 1 - 1.35iT - 41T^{2} \)
43 \( 1 + (-3.16 + 3.16i)T - 43iT^{2} \)
47 \( 1 + (8.91 - 8.91i)T - 47iT^{2} \)
53 \( 1 + (5.71 + 5.71i)T + 53iT^{2} \)
59 \( 1 + 11.6T + 59T^{2} \)
61 \( 1 + 11.6T + 61T^{2} \)
67 \( 1 + (2.64 + 2.64i)T + 67iT^{2} \)
71 \( 1 - 14.9iT - 71T^{2} \)
73 \( 1 + (1.34 - 1.34i)T - 73iT^{2} \)
79 \( 1 - 3.30iT - 79T^{2} \)
83 \( 1 + (10.0 + 10.0i)T + 83iT^{2} \)
89 \( 1 - 9.24T + 89T^{2} \)
97 \( 1 + (6.16 + 6.16i)T + 97iT^{2} \)
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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

−9.037508881939333607504487519012, −8.276771644163621545491248686000, −7.67732891860964581580907688819, −6.79320807267578723165921889689, −5.97639479611423948179510105393, −4.97452439067357186004948076589, −4.17965381037053029802794589384, −3.07801286622324550270576265436, −1.38673339939655010269292542932, −0.20745056691644464429303433124, 1.63029188387829871658015117765, 2.88742004912361804263436760129, 3.55617773317696747067917515004, 4.51370087019465319360470445876, 6.03541028369398579313714474024, 6.52116566338415105704615724019, 7.55292054304187253676503311219, 8.269392802491036018697568349132, 9.058847479909618841004106929386, 9.783009130223819750732699510742

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