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Properties

Label 2-1530-15.2-c1-0-26
Degree $2$
Conductor $1530$
Sign $-0.662 + 0.749i$
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 + (0.707 − 2.12i)5-s + (−3 − 3i)7-s + (0.707 + 0.707i)8-s + (0.999 + 2i)10-s − 2.82i·11-s + (4 − 4i)13-s + 4.24·14-s − 1.00·16-s + (0.707 − 0.707i)17-s + 4i·19-s + (−2.12 − 0.707i)20-s + (2.00 + 2.00i)22-s + (4.24 + 4.24i)23-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (0.316 − 0.948i)5-s + (−1.13 − 1.13i)7-s + (0.250 + 0.250i)8-s + (0.316 + 0.632i)10-s − 0.852i·11-s + (1.10 − 1.10i)13-s + 1.13·14-s − 0.250·16-s + (0.171 − 0.171i)17-s + 0.917i·19-s + (−0.474 − 0.158i)20-s + (0.426 + 0.426i)22-s + (0.884 + 0.884i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1530\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 17\)
Sign: $-0.662 + 0.749i$
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: \((2,\ 1530,\ (\ :1/2),\ -0.662 + 0.749i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8940489550\)
\(L(\frac12)\) \(\approx\) \(0.8940489550\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 + 2.12i)T \)
17 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 + (3 + 3i)T + 7iT^{2} \)
11 \( 1 + 2.82iT - 11T^{2} \)
13 \( 1 + (-4 + 4i)T - 13iT^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + (-4.24 - 4.24i)T + 23iT^{2} \)
29 \( 1 - 4.24T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + (5 + 5i)T + 37iT^{2} \)
41 \( 1 + 8.48iT - 41T^{2} \)
43 \( 1 + (3 - 3i)T - 43iT^{2} \)
47 \( 1 + (-2.82 + 2.82i)T - 47iT^{2} \)
53 \( 1 + (1.41 + 1.41i)T + 53iT^{2} \)
59 \( 1 - 9.89T + 59T^{2} \)
61 \( 1 + 14T + 61T^{2} \)
67 \( 1 + (5 + 5i)T + 67iT^{2} \)
71 \( 1 - 15.5iT - 71T^{2} \)
73 \( 1 + (-6 + 6i)T - 73iT^{2} \)
79 \( 1 - 10iT - 79T^{2} \)
83 \( 1 + (4.24 + 4.24i)T + 83iT^{2} \)
89 \( 1 + 18.3T + 89T^{2} \)
97 \( 1 + 97iT^{2} \)
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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

−9.061098925698761600016957207321, −8.455647211085013899241056123360, −7.61641382040056977208647237765, −6.80573570574302096051973011109, −5.79950962349006006230640952357, −5.43206521573945811422444974355, −3.93963764441495897922141958696, −3.28402873597875441593004291403, −1.35082024844423673701306853053, −0.44021519579004013850734104351, 1.74348732493561622527155622845, 2.71847536393846975734468998404, 3.38790231179795205392368173984, 4.62310972595768800043695858365, 5.93040645607088074411801945901, 6.66027790492998689600346248798, 7.11475808991024722711369655772, 8.497811028326530402807892427569, 9.126669463990325302515806208918, 9.646446310948944634740613588061

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