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Properties

Label 2-1530-17.13-c1-0-5
Degree $2$
Conductor $1530$
Sign $0.914 - 0.405i$
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  − i·2-s − 4-s + (−0.707 − 0.707i)5-s + (−1.84 + 1.84i)7-s + i·8-s + (−0.707 + 0.707i)10-s + (1.19 − 1.19i)11-s − 2.72·13-s + (1.84 + 1.84i)14-s + 16-s + (−1.19 − 3.94i)17-s + 0.663i·19-s + (0.707 + 0.707i)20-s + (−1.19 − 1.19i)22-s + (0.249 − 0.249i)23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (−0.316 − 0.316i)5-s + (−0.698 + 0.698i)7-s + 0.353i·8-s + (−0.223 + 0.223i)10-s + (0.361 − 0.361i)11-s − 0.757·13-s + (0.493 + 0.493i)14-s + 0.250·16-s + (−0.290 − 0.956i)17-s + 0.152i·19-s + (0.158 + 0.158i)20-s + (−0.255 − 0.255i)22-s + (0.0520 − 0.0520i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 - 0.405i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.914 - 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1530\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 17\)
Sign: $0.914 - 0.405i$
Analytic conductor: \(12.2171\)
Root analytic conductor: \(3.49529\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1530} (1441, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1530,\ (\ :1/2),\ 0.914 - 0.405i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9835345730\)
\(L(\frac12)\) \(\approx\) \(0.9835345730\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
5 \( 1 + (0.707 + 0.707i)T \)
17 \( 1 + (1.19 + 3.94i)T \)
good7 \( 1 + (1.84 - 1.84i)T - 7iT^{2} \)
11 \( 1 + (-1.19 + 1.19i)T - 11iT^{2} \)
13 \( 1 + 2.72T + 13T^{2} \)
19 \( 1 - 0.663iT - 19T^{2} \)
23 \( 1 + (-0.249 + 0.249i)T - 23iT^{2} \)
29 \( 1 + (-5.08 - 5.08i)T + 29iT^{2} \)
31 \( 1 + (-6.72 - 6.72i)T + 31iT^{2} \)
37 \( 1 + (-0.352 - 0.352i)T + 37iT^{2} \)
41 \( 1 + (5.42 - 5.42i)T - 41iT^{2} \)
43 \( 1 - 9.41iT - 43T^{2} \)
47 \( 1 - 3.33T + 47T^{2} \)
53 \( 1 - 5.06iT - 53T^{2} \)
59 \( 1 - 10.7iT - 59T^{2} \)
61 \( 1 + (-5.19 + 5.19i)T - 61iT^{2} \)
67 \( 1 - 3.14T + 67T^{2} \)
71 \( 1 + (1.21 + 1.21i)T + 71iT^{2} \)
73 \( 1 + (-10.6 - 10.6i)T + 73iT^{2} \)
79 \( 1 + (-1.85 + 1.85i)T - 79iT^{2} \)
83 \( 1 + 11.3iT - 83T^{2} \)
89 \( 1 - 5.25T + 89T^{2} \)
97 \( 1 + (8.43 + 8.43i)T + 97iT^{2} \)
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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

−9.539562201198121450693662318536, −8.859478488754718684961799221582, −8.196651727837656661039246716008, −7.05766354335658755931785570896, −6.25340802758380644897518657170, −5.13658345418725397457279927592, −4.49935682346536171517261521750, −3.20919754817757740732585806421, −2.64144504135974535192217595026, −1.09392510246202154722462320612, 0.45633245651796772903104632940, 2.32709401789034669755801156711, 3.68506173559809493195937186956, 4.27378533619878348615122835165, 5.34351182189747606065061686301, 6.53497631429318734598524156162, 6.77685829888457443429287038423, 7.75964930037614030201847501668, 8.395632234626752340136336085772, 9.474479413379487167764428846338

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