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Properties

Label 2-1530-15.2-c1-0-29
Degree $2$
Conductor $1530$
Sign $-0.998 + 0.0618i$
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 − i)5-s + (−3.41 − 3.41i)7-s + (−0.707 − 0.707i)8-s + (0.707 − 2.12i)10-s + 2.82i·11-s + (−1.41 + 1.41i)13-s − 4.82·14-s − 1.00·16-s + (−0.707 + 0.707i)17-s − 4.82i·19-s + (−1.00 − 2.00i)20-s + (2.00 + 2.00i)22-s + (−6.41 − 6.41i)23-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s − 0.500i·4-s + (0.894 − 0.447i)5-s + (−1.29 − 1.29i)7-s + (−0.250 − 0.250i)8-s + (0.223 − 0.670i)10-s + 0.852i·11-s + (−0.392 + 0.392i)13-s − 1.29·14-s − 0.250·16-s + (−0.171 + 0.171i)17-s − 1.10i·19-s + (−0.223 − 0.447i)20-s + (0.426 + 0.426i)22-s + (−1.33 − 1.33i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0618i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1530\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 17\)
Sign: $-0.998 + 0.0618i$
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: \((2,\ 1530,\ (\ :1/2),\ -0.998 + 0.0618i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.391532190\)
\(L(\frac12)\) \(\approx\) \(1.391532190\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 + i)T \)
17 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 + (3.41 + 3.41i)T + 7iT^{2} \)
11 \( 1 - 2.82iT - 11T^{2} \)
13 \( 1 + (1.41 - 1.41i)T - 13iT^{2} \)
19 \( 1 + 4.82iT - 19T^{2} \)
23 \( 1 + (6.41 + 6.41i)T + 23iT^{2} \)
29 \( 1 - 3.65T + 29T^{2} \)
31 \( 1 + 9.07T + 31T^{2} \)
37 \( 1 + (-4 - 4i)T + 37iT^{2} \)
41 \( 1 + 3.17iT - 41T^{2} \)
43 \( 1 + (1.58 - 1.58i)T - 43iT^{2} \)
47 \( 1 - 47iT^{2} \)
53 \( 1 + (8.24 + 8.24i)T + 53iT^{2} \)
59 \( 1 - 14.7T + 59T^{2} \)
61 \( 1 - 5.41T + 61T^{2} \)
67 \( 1 + (2.41 + 2.41i)T + 67iT^{2} \)
71 \( 1 - 5.65iT - 71T^{2} \)
73 \( 1 + (8.24 - 8.24i)T - 73iT^{2} \)
79 \( 1 - 3.41iT - 79T^{2} \)
83 \( 1 + (6.82 + 6.82i)T + 83iT^{2} \)
89 \( 1 + 5.41T + 89T^{2} \)
97 \( 1 + (1.41 + 1.41i)T + 97iT^{2} \)
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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

−9.471724580116937760346720688355, −8.446727218732176882114812472830, −7.01172691999377704377146055562, −6.72755488254138075874541688070, −5.74901580376631040561569892810, −4.61229391248232579815568020684, −4.10846903477143880261385920983, −2.86467209996394140732691630198, −1.88560758827843434413358567914, −0.41969389055494805698019521706, 2.07667949689220683687718464455, 3.02117531289024341528830006066, 3.71154381009074207041450757343, 5.36569141981234446660685527402, 5.84212794904990270176658496441, 6.25640295654233746745212902324, 7.27441623660568645403810832408, 8.230553293945744516445323613613, 9.171293891971254753733744481144, 9.661833945949261442628239415796

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