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Properties

Label 2-1470-105.104-c1-0-13
Degree $2$
Conductor $1470$
Sign $0.207 - 0.978i$
Analytic cond. $11.7380$
Root an. cond. $3.42607$
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  − 2-s + (1.18 − 1.26i)3-s + 4-s + (−0.686 + 2.12i)5-s + (−1.18 + 1.26i)6-s − 8-s + (−0.186 − 2.99i)9-s + (0.686 − 2.12i)10-s + 4.25i·11-s + (1.18 − 1.26i)12-s + 2·13-s + (1.87 + 3.39i)15-s + 16-s + 6.63i·17-s + (0.186 + 2.99i)18-s − 3.46i·19-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.684 − 0.728i)3-s + 0.5·4-s + (−0.306 + 0.951i)5-s + (−0.484 + 0.515i)6-s − 0.353·8-s + (−0.0620 − 0.998i)9-s + (0.216 − 0.672i)10-s + 1.28i·11-s + (0.342 − 0.364i)12-s + 0.554·13-s + (0.483 + 0.875i)15-s + 0.250·16-s + 1.60i·17-s + (0.0438 + 0.705i)18-s − 0.794i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.207 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.207 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.207 - 0.978i$
Analytic conductor: \(11.7380\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (1469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ 0.207 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.120311441\)
\(L(\frac12)\) \(\approx\) \(1.120311441\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (-1.18 + 1.26i)T \)
5 \( 1 + (0.686 - 2.12i)T \)
7 \( 1 \)
good11 \( 1 - 4.25iT - 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 - 6.63iT - 17T^{2} \)
19 \( 1 + 3.46iT - 19T^{2} \)
23 \( 1 + 4.37T + 23T^{2} \)
29 \( 1 + 3.31iT - 29T^{2} \)
31 \( 1 - 2.37iT - 31T^{2} \)
37 \( 1 - 11.6iT - 37T^{2} \)
41 \( 1 + 1.62T + 41T^{2} \)
43 \( 1 - 11.0iT - 43T^{2} \)
47 \( 1 - 1.87iT - 47T^{2} \)
53 \( 1 + 1.37T + 53T^{2} \)
59 \( 1 - 4.11T + 59T^{2} \)
61 \( 1 + 2.81iT - 61T^{2} \)
67 \( 1 - 7.57iT - 67T^{2} \)
71 \( 1 - 1.87iT - 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + 8.11T + 79T^{2} \)
83 \( 1 - 1.43iT - 83T^{2} \)
89 \( 1 - 4.37T + 89T^{2} \)
97 \( 1 + 2.11T + 97T^{2} \)
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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

−9.805984874064418165735722263403, −8.654119781839971927332961760458, −8.065483280521035871374646829898, −7.41649936141942260557022385822, −6.57668605501284717703182103738, −6.15583460359966878152111982403, −4.37554201953111623964193930691, −3.38669076185055253976730381443, −2.39686207906888075738662764130, −1.49403882864353223831208306246, 0.51531857712000973520248554613, 2.01987623157398293807971071433, 3.31266887063717673428892783460, 4.01629380144446891755594333675, 5.22415974450616130168740444406, 5.87664325928540381586278541670, 7.31464031587864819396760719728, 7.997626322832261099491364757844, 8.752056099614941180277948529393, 9.081335637348352091827789512859

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