Properties

Degree $2$
Conductor $1470$
Sign $0.730 + 0.683i$
Motivic weight $1$
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 + (1.63 + 1.52i)5-s + 6-s + i·8-s − 9-s + (1.52 − 1.63i)10-s + 4.46·11-s i·12-s − 5.88i·13-s + (−1.52 + 1.63i)15-s + 16-s − 7.73i·17-s + i·18-s − 6.61·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (0.730 + 0.683i)5-s + 0.408·6-s + 0.353i·8-s − 0.333·9-s + (0.483 − 0.516i)10-s + 1.34·11-s − 0.288i·12-s − 1.63i·13-s + (−0.394 + 0.421i)15-s + 0.250·16-s − 1.87i·17-s + 0.235i·18-s − 1.51·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.730 + 0.683i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.730 + 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.730 + 0.683i$
Motivic weight: \(1\)
Character: $\chi_{1470} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ 0.730 + 0.683i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.891042819\)
\(L(\frac12)\) \(\approx\) \(1.891042819\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.63 - 1.52i)T \)
7 \( 1 \)
good11 \( 1 - 4.46T + 11T^{2} \)
13 \( 1 + 5.88iT - 13T^{2} \)
17 \( 1 + 7.73iT - 17T^{2} \)
19 \( 1 + 6.61T + 19T^{2} \)
23 \( 1 + 2.61iT - 23T^{2} \)
29 \( 1 - 8.17T + 29T^{2} \)
31 \( 1 - 8.46T + 31T^{2} \)
37 \( 1 - 3.18iT - 37T^{2} \)
41 \( 1 - 3.56T + 41T^{2} \)
43 \( 1 + 1.43iT - 43T^{2} \)
47 \( 1 - 6.79iT - 47T^{2} \)
53 \( 1 - 1.38iT - 53T^{2} \)
59 \( 1 + 4.66T + 59T^{2} \)
61 \( 1 - 9.79T + 61T^{2} \)
67 \( 1 - 1.85iT - 67T^{2} \)
71 \( 1 - 2.02T + 71T^{2} \)
73 \( 1 + 4.01iT - 73T^{2} \)
79 \( 1 - 6.98T + 79T^{2} \)
83 \( 1 + 5.35iT - 83T^{2} \)
89 \( 1 + 14.3T + 89T^{2} \)
97 \( 1 + 7.71iT - 97T^{2} \)
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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

−9.630236878958459901756942912405, −8.873671705653559392261885719970, −8.045534560758007332774049989327, −6.76559418768904874096708750348, −6.13869100192409424353587114002, −5.05570125460045841422964773894, −4.27214720632854756794314729677, −3.04917030068978204318999567876, −2.55523549639596507158647859901, −0.889614785385598890267374279983, 1.26463438058089183687511714733, 2.10684397514615877681710765312, 4.00808344906820659418692543248, 4.49437298933034487981161617281, 5.81999939670580109944034736216, 6.52794758005241517385137722900, 6.73217744905807100863378592020, 8.330170290417244177754172124993, 8.531160516898648642003364764786, 9.363903847767506236693879513681

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