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Properties

Label 2-1470-35.9-c1-0-19
Degree $2$
Conductor $1470$
Sign $0.999 - 0.0277i$
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  + (0.866 + 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + (−2.20 + 0.344i)5-s − 0.999·6-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (−2.08 − 0.806i)10-s + (−0.838 − 1.45i)11-s + (−0.866 − 0.499i)12-s − 4.48i·13-s + (1.74 − 1.40i)15-s + (−0.5 + 0.866i)16-s + (6.59 − 3.80i)17-s + (0.866 − 0.499i)18-s + (−2.24 + 3.88i)19-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.499 + 0.288i)3-s + (0.249 + 0.433i)4-s + (−0.988 + 0.153i)5-s − 0.408·6-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.659 − 0.255i)10-s + (−0.252 − 0.437i)11-s + (−0.249 − 0.144i)12-s − 1.24i·13-s + (0.449 − 0.362i)15-s + (−0.125 + 0.216i)16-s + (1.59 − 0.922i)17-s + (0.204 − 0.117i)18-s + (−0.514 + 0.890i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0277i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0277i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.999 - 0.0277i$
Analytic conductor: \(11.7380\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ 0.999 - 0.0277i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.555960748\)
\(L(\frac12)\) \(\approx\) \(1.555960748\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (2.20 - 0.344i)T \)
7 \( 1 \)
good11 \( 1 + (0.838 + 1.45i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.48iT - 13T^{2} \)
17 \( 1 + (-6.59 + 3.80i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.24 - 3.88i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.417 - 0.241i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.19T + 29T^{2} \)
31 \( 1 + (1.74 + 3.01i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.417 + 0.241i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 12.0T + 41T^{2} \)
43 \( 1 + 2iT - 43T^{2} \)
47 \( 1 + (-9.91 - 5.72i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-8.29 + 4.78i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.88 + 4.99i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.28 - 9.15i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.90 + 1.67i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (3.46 - 2i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.25 + 3.91i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.87iT - 83T^{2} \)
89 \( 1 + (-1.67 + 2.90i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 17.7iT - 97T^{2} \)
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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

−9.593431399852529114574966088256, −8.425350487913286699780032477314, −7.70123998153529931892748097995, −7.23668443515838481738124391771, −5.86544918316085695514523141593, −5.55396455627021173151779897415, −4.44437021300707166743132901244, −3.59841167215632072762742098007, −2.83735052436357499420225343532, −0.67800308266051386694389992094, 1.06804378506856054943691874530, 2.37199938606623105161826565550, 3.68751801008880766826689601014, 4.36530407683471054200753747618, 5.22008733507858594030419273059, 6.15952952796657163005182406471, 7.08301200170237352005931541613, 7.65729685899979061065571910111, 8.702396142367983758528264833716, 9.579653389513032064792751739054

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