Properties

Degree $2$
Conductor $1470$
Sign $-0.844 - 0.535i$
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 + (−1.88 − 1.19i)5-s + 6-s i·8-s − 9-s + (1.19 − 1.88i)10-s + 0.979·11-s + i·12-s − 0.435i·13-s + (−1.19 + 1.88i)15-s + 16-s − 2.79i·17-s i·18-s − 7.34·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (−0.844 − 0.535i)5-s + 0.408·6-s − 0.353i·8-s − 0.333·9-s + (0.378 − 0.597i)10-s + 0.295·11-s + 0.288i·12-s − 0.120i·13-s + (−0.308 + 0.487i)15-s + 0.250·16-s − 0.678i·17-s − 0.235i·18-s − 1.68·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.844 - 0.535i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.844 - 0.535i$
Motivic weight: \(1\)
Character: $\chi_{1470} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ -0.844 - 0.535i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3338059448\)
\(L(\frac12)\) \(\approx\) \(0.3338059448\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.88 + 1.19i)T \)
7 \( 1 \)
good11 \( 1 - 0.979T + 11T^{2} \)
13 \( 1 + 0.435iT - 13T^{2} \)
17 \( 1 + 2.79iT - 17T^{2} \)
19 \( 1 + 7.34T + 19T^{2} \)
23 \( 1 - 3.34iT - 23T^{2} \)
29 \( 1 + 3.74T + 29T^{2} \)
31 \( 1 - 4.97T + 31T^{2} \)
37 \( 1 - 4.63iT - 37T^{2} \)
41 \( 1 - 4.94T + 41T^{2} \)
43 \( 1 - 9.97iT - 43T^{2} \)
47 \( 1 - 4.40iT - 47T^{2} \)
53 \( 1 + 0.657iT - 53T^{2} \)
59 \( 1 + 8.27T + 59T^{2} \)
61 \( 1 + 11.3T + 61T^{2} \)
67 \( 1 - 2.36iT - 67T^{2} \)
71 \( 1 + 14.1T + 71T^{2} \)
73 \( 1 - 15.3iT - 73T^{2} \)
79 \( 1 - 2.88T + 79T^{2} \)
83 \( 1 + 14.3iT - 83T^{2} \)
89 \( 1 + 10.1T + 89T^{2} \)
97 \( 1 - 10.6iT - 97T^{2} \)
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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

−9.480311098494418170819561470387, −8.850095676508639626002879453641, −8.035192105498530390703168863124, −7.55422233053971612677231084585, −6.63428213395507846378952752440, −5.91781399442292225724029392832, −4.80098034999287935500757220097, −4.14142726978107382616959459329, −2.91316819571186806665311956821, −1.29731367126722137171792257794, 0.14142492717513663951394777249, 2.04487790930792894240492300162, 3.12262590042375763855506076473, 4.09349286404083913769938130011, 4.47061488995238942930143552352, 5.85181559537170711300073837642, 6.69228908550568418527964503536, 7.76879658862145729023696763672, 8.577390883301794966768927755810, 9.151279338026816737608159102988

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