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Properties

Label 2-1470-7.2-c1-0-13
Degree $2$
Conductor $1470$
Sign $0.827 + 0.561i$
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  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)5-s + 0.999·6-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (−0.499 − 0.866i)10-s + (−0.292 − 0.507i)11-s + (−0.499 + 0.866i)12-s + 0.999·15-s + (−0.5 + 0.866i)16-s + (−0.707 − 1.22i)17-s + (−0.499 − 0.866i)18-s + (1.41 − 2.44i)19-s + 0.999·20-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.223 + 0.387i)5-s + 0.408·6-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.158 − 0.273i)10-s + (−0.0883 − 0.152i)11-s + (−0.144 + 0.249i)12-s + 0.258·15-s + (−0.125 + 0.216i)16-s + (−0.171 − 0.297i)17-s + (−0.117 − 0.204i)18-s + (0.324 − 0.561i)19-s + 0.223·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.827 + 0.561i$
Analytic conductor: \(11.7380\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ 0.827 + 0.561i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9460392044\)
\(L(\frac12)\) \(\approx\) \(0.9460392044\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 \)
good11 \( 1 + (0.292 + 0.507i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (0.707 + 1.22i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.41 + 2.44i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.41 - 4.18i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 8.24T + 29T^{2} \)
31 \( 1 + (2.53 + 4.39i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.707 + 1.22i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.82T + 41T^{2} \)
43 \( 1 - 4.58T + 43T^{2} \)
47 \( 1 + (-4.53 + 7.85i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.65 - 8.06i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.24 + 2.15i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.82 + 10.0i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.94 + 6.84i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.8T + 71T^{2} \)
73 \( 1 + (3.82 + 6.63i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.65 + 9.79i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.17T + 83T^{2} \)
89 \( 1 + (-3.24 + 5.61i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 8.82T + 97T^{2} \)
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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

−9.324556439356993142740770260768, −8.476756416243194493551483074659, −7.68718906640570437126550726620, −7.06936727255151821341352600517, −6.31224961022658230080042850929, −5.50061450971152352497789218676, −4.57712114761661048296297274895, −3.32787646149041249381882683378, −2.06023741250319486092324017762, −0.54717203242586045290212328605, 1.03625650107283008920231458933, 2.44675397965694623883318618618, 3.59804297133878023242134720295, 4.41639242852346952976546110091, 5.22769316478609199825667658685, 6.28423519298494638676699618511, 7.27315649227564570732588613967, 8.359129533893921444274754838059, 8.723346961336526383944148963640, 9.810603933507813314988084904747

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