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Properties

Label 2-1470-5.4-c1-0-31
Degree $2$
Conductor $1470$
Sign $0.988 + 0.153i$
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  + i·2-s − i·3-s − 4-s + (2.20 + 0.344i)5-s + 6-s − i·8-s − 9-s + (−0.344 + 2.20i)10-s + 6.10·11-s + i·12-s − 1.68i·13-s + (0.344 − 2.20i)15-s + 16-s − 6.83i·17-s − i·18-s − 1.68·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (0.988 + 0.153i)5-s + 0.408·6-s − 0.353i·8-s − 0.333·9-s + (−0.108 + 0.698i)10-s + 1.84·11-s + 0.288i·12-s − 0.468i·13-s + (0.0888 − 0.570i)15-s + 0.250·16-s − 1.65i·17-s − 0.235i·18-s − 0.387·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.153i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.988 + 0.153i$
Analytic conductor: \(11.7380\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ 0.988 + 0.153i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.041407587\)
\(L(\frac12)\) \(\approx\) \(2.041407587\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
5 \( 1 + (-2.20 - 0.344i)T \)
7 \( 1 \)
good11 \( 1 - 6.10T + 11T^{2} \)
13 \( 1 + 1.68iT - 13T^{2} \)
17 \( 1 + 6.83iT - 17T^{2} \)
19 \( 1 + 1.68T + 19T^{2} \)
23 \( 1 + 2.31iT - 23T^{2} \)
29 \( 1 + 8.41T + 29T^{2} \)
31 \( 1 + 0.688T + 31T^{2} \)
37 \( 1 - 2.31iT - 37T^{2} \)
41 \( 1 - 5.14T + 41T^{2} \)
43 \( 1 - 2iT - 43T^{2} \)
47 \( 1 + 3.06iT - 47T^{2} \)
53 \( 1 - 10.4iT - 53T^{2} \)
59 \( 1 - 7.04T + 59T^{2} \)
61 \( 1 - 9.46T + 61T^{2} \)
67 \( 1 + 12.2iT - 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + 4iT - 73T^{2} \)
79 \( 1 + 7.31T + 79T^{2} \)
83 \( 1 + 13.5iT - 83T^{2} \)
89 \( 1 - 12.2T + 89T^{2} \)
97 \( 1 + 4.95iT - 97T^{2} \)
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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

−9.215846798756732720917060526955, −8.883034174621803467526254694527, −7.64529734333378639001368343480, −6.94708377955578145196020666715, −6.32465927958963145893873200008, −5.62213793622099385413643862439, −4.65010320904731160907007833926, −3.44136828798755727750582068112, −2.18610881993371807543755290364, −0.931331034842372190810524805246, 1.37956142411026077863953869654, 2.18813479494393730126419119963, 3.76803277727488799508832031825, 4.06461009041190486666230887583, 5.36757173415166720928703609958, 6.08866404930733228351735978490, 6.90707313741129575281615051361, 8.364608289140266582748296356225, 9.021919677446729966961268642503, 9.556336681336712510444534944731

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