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Properties

Label 2-1530-255.98-c1-0-23
Degree $2$
Conductor $1530$
Sign $0.642 + 0.766i$
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 + 0.0740i)5-s + 0.956i·7-s + (0.707 + 0.707i)8-s + (−1.63 + 1.52i)10-s + (−1.88 − 1.88i)11-s + (−1.95 − 1.95i)13-s + (−0.676 − 0.676i)14-s − 1.00·16-s + (−0.844 − 4.03i)17-s + 1.26·19-s + (0.0740 − 2.23i)20-s + 2.66·22-s − 6.61i·23-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (0.999 + 0.0330i)5-s + 0.361i·7-s + (0.250 + 0.250i)8-s + (−0.516 + 0.483i)10-s + (−0.567 − 0.567i)11-s + (−0.541 − 0.541i)13-s + (−0.180 − 0.180i)14-s − 0.250·16-s + (−0.204 − 0.978i)17-s + 0.290·19-s + (0.0165 − 0.499i)20-s + 0.567·22-s − 1.37i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1530\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 17\)
Sign: $0.642 + 0.766i$
Analytic conductor: \(12.2171\)
Root analytic conductor: \(3.49529\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1530} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1530,\ (\ :1/2),\ 0.642 + 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.210632174\)
\(L(\frac12)\) \(\approx\) \(1.210632174\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.23 - 0.0740i)T \)
17 \( 1 + (0.844 + 4.03i)T \)
good7 \( 1 - 0.956iT - 7T^{2} \)
11 \( 1 + (1.88 + 1.88i)T + 11iT^{2} \)
13 \( 1 + (1.95 + 1.95i)T + 13iT^{2} \)
19 \( 1 - 1.26T + 19T^{2} \)
23 \( 1 + 6.61iT - 23T^{2} \)
29 \( 1 + (2.53 + 2.53i)T + 29iT^{2} \)
31 \( 1 + (2.95 + 2.95i)T + 31iT^{2} \)
37 \( 1 - 7.53T + 37T^{2} \)
41 \( 1 + (-1.63 - 1.63i)T + 41iT^{2} \)
43 \( 1 + (1.45 - 1.45i)T - 43iT^{2} \)
47 \( 1 + (4.83 + 4.83i)T + 47iT^{2} \)
53 \( 1 + (7.57 + 7.57i)T + 53iT^{2} \)
59 \( 1 - 2.87iT - 59T^{2} \)
61 \( 1 + (5.15 - 5.15i)T - 61iT^{2} \)
67 \( 1 + (-2.85 + 2.85i)T - 67iT^{2} \)
71 \( 1 + (-2.91 + 2.91i)T - 71iT^{2} \)
73 \( 1 - 4.43iT - 73T^{2} \)
79 \( 1 + (-5.16 - 5.16i)T + 79iT^{2} \)
83 \( 1 + (2.66 + 2.66i)T + 83iT^{2} \)
89 \( 1 - 2.91T + 89T^{2} \)
97 \( 1 - 14.9T + 97T^{2} \)
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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

−9.425660973667727514490214045174, −8.558172688160068130160400129297, −7.81903785306339471035260943906, −6.92059056615197966419343736990, −6.06243112242596488086244060807, −5.43327818010875426132436283056, −4.64454350042499428597497997568, −2.94467316032228661640586955716, −2.18007877121387231807931352457, −0.55960834019496467117104469884, 1.42693736164628703566957946424, 2.23126642382742291346444612646, 3.37705386599773684510112040514, 4.51990160789925086849029574326, 5.42172842041624542330369068420, 6.39871678869235886469270106815, 7.32238312878368226231651192844, 7.957272735845199976100084197623, 9.192349149769184610371460488778, 9.454473056831607561820032100650

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