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Properties

Label 2-1530-15.2-c1-0-24
Degree $2$
Conductor $1530$
Sign $-0.941 + 0.335i$
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 + (−1.64 + 1.51i)5-s + (0.552 + 0.552i)7-s + (−0.707 − 0.707i)8-s + (−0.0917 + 2.23i)10-s − 6.31i·11-s + (−2.87 + 2.87i)13-s + 0.781·14-s − 1.00·16-s + (0.707 − 0.707i)17-s + 0.819i·19-s + (1.51 + 1.64i)20-s + (−4.46 − 4.46i)22-s + (−1.74 − 1.74i)23-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s − 0.500i·4-s + (−0.735 + 0.677i)5-s + (0.208 + 0.208i)7-s + (−0.250 − 0.250i)8-s + (−0.0290 + 0.706i)10-s − 1.90i·11-s + (−0.798 + 0.798i)13-s + 0.208·14-s − 0.250·16-s + (0.171 − 0.171i)17-s + 0.188i·19-s + (0.338 + 0.367i)20-s + (−0.952 − 0.952i)22-s + (−0.363 − 0.363i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.941 + 0.335i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.941 + 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1530\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 17\)
Sign: $-0.941 + 0.335i$
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: \((2,\ 1530,\ (\ :1/2),\ -0.941 + 0.335i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8771613181\)
\(L(\frac12)\) \(\approx\) \(0.8771613181\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.64 - 1.51i)T \)
17 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 + (-0.552 - 0.552i)T + 7iT^{2} \)
11 \( 1 + 6.31iT - 11T^{2} \)
13 \( 1 + (2.87 - 2.87i)T - 13iT^{2} \)
19 \( 1 - 0.819iT - 19T^{2} \)
23 \( 1 + (1.74 + 1.74i)T + 23iT^{2} \)
29 \( 1 - 4.59T + 29T^{2} \)
31 \( 1 + 8.39T + 31T^{2} \)
37 \( 1 + (5.24 + 5.24i)T + 37iT^{2} \)
41 \( 1 + 9.72iT - 41T^{2} \)
43 \( 1 + (6.34 - 6.34i)T - 43iT^{2} \)
47 \( 1 + (0.839 - 0.839i)T - 47iT^{2} \)
53 \( 1 + (2.65 + 2.65i)T + 53iT^{2} \)
59 \( 1 + 9.73T + 59T^{2} \)
61 \( 1 - 12.1T + 61T^{2} \)
67 \( 1 + (11.2 + 11.2i)T + 67iT^{2} \)
71 \( 1 + 10.9iT - 71T^{2} \)
73 \( 1 + (2.69 - 2.69i)T - 73iT^{2} \)
79 \( 1 - 16.7iT - 79T^{2} \)
83 \( 1 + (6.72 + 6.72i)T + 83iT^{2} \)
89 \( 1 - 7.62T + 89T^{2} \)
97 \( 1 + (-3.34 - 3.34i)T + 97iT^{2} \)
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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

−9.052789955310651880518445358071, −8.393184426583082242736146580182, −7.44078496969582571233208999073, −6.58418637411372968494655456706, −5.76047322263797609382476960053, −4.82664930229642305287687487093, −3.74434934102556829756785265909, −3.16076773240056433522548188801, −2.03500215929573735398783765666, −0.27905802641448652591275450001, 1.66263423329599025255632445902, 3.06835639333093332641294405317, 4.21018666077908859198161322386, 4.79795084331044795686474944330, 5.44930694845389735716212738056, 6.77855881202795887290561194647, 7.46106261252600476160021657973, 7.927343506621112239952745406440, 8.877325153267403620024109376288, 9.788725150360485093107238441474

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