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Properties

Label 1-1480-1480.1197-r0-0-0
Degree $1$
Conductor $1480$
Sign $-0.724 - 0.689i$
Analytic cond. $6.87309$
Root an. cond. $6.87309$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.342 + 0.939i)3-s + (0.984 + 0.173i)7-s + (−0.766 + 0.642i)9-s + (−0.5 − 0.866i)11-s + (−0.766 − 0.642i)13-s + (−0.766 + 0.642i)17-s + (−0.342 − 0.939i)19-s + (0.173 + 0.984i)21-s + (−0.5 + 0.866i)23-s + (−0.866 − 0.5i)27-s + (−0.866 + 0.5i)29-s − i·31-s + (0.642 − 0.766i)33-s + (0.342 − 0.939i)39-s + (−0.766 − 0.642i)41-s + ⋯
L(s)  = 1  + (0.342 + 0.939i)3-s + (0.984 + 0.173i)7-s + (−0.766 + 0.642i)9-s + (−0.5 − 0.866i)11-s + (−0.766 − 0.642i)13-s + (−0.766 + 0.642i)17-s + (−0.342 − 0.939i)19-s + (0.173 + 0.984i)21-s + (−0.5 + 0.866i)23-s + (−0.866 − 0.5i)27-s + (−0.866 + 0.5i)29-s − i·31-s + (0.642 − 0.766i)33-s + (0.342 − 0.939i)39-s + (−0.766 − 0.642i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.724 - 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.724 - 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $-0.724 - 0.689i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (1197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ -0.724 - 0.689i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001413030642 + 0.003537174166i\)
\(L(\frac12)\) \(\approx\) \(0.001413030642 + 0.003537174166i\)
\(L(1)\) \(\approx\) \(0.8554239768 + 0.2318121591i\)
\(L(1)\) \(\approx\) \(0.8554239768 + 0.2318121591i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
37 \( 1 \)
good3 \( 1 + (0.342 + 0.939i)T \)
7 \( 1 + (0.984 + 0.173i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.766 - 0.642i)T \)
17 \( 1 + (-0.766 + 0.642i)T \)
19 \( 1 + (-0.342 - 0.939i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.866 + 0.5i)T \)
31 \( 1 - iT \)
41 \( 1 + (-0.766 - 0.642i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.866 - 0.5i)T \)
53 \( 1 + (-0.984 + 0.173i)T \)
59 \( 1 + (-0.984 + 0.173i)T \)
61 \( 1 + (-0.642 + 0.766i)T \)
67 \( 1 + (0.984 + 0.173i)T \)
71 \( 1 + (-0.939 + 0.342i)T \)
73 \( 1 + iT \)
79 \( 1 + (-0.984 - 0.173i)T \)
83 \( 1 + (0.642 + 0.766i)T \)
89 \( 1 + (-0.984 + 0.173i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.29881284615195302613836723246, −19.46034887366167241691631438647, −18.48363903637913384039450050864, −18.14984702790370573378571128524, −17.3176901286067158095093853512, −16.6753601664500961762791917128, −15.48120155546424224075771412916, −14.63138955251261685690497668738, −14.25463589870982393491083983839, −13.36566382221314049525363928381, −12.5351717429391911481888850482, −11.9226626612029151275961438219, −11.1752330874987973507089885985, −10.15322263193061713597029939507, −9.2742434554354328979429455195, −8.317124127947698664253670358135, −7.7425967617049249835894269046, −7.00241265945381225346627393080, −6.24619109062534307932549523760, −5.00024230163162017632887818211, −4.43633888185202492869137136900, −3.11928246072135484740671056897, −1.99182392665657316568619442902, −1.71678997004634577864562764304, −0.00118375695297733482488452508, 1.80148380329985394123079426838, 2.66805769308370379484276089028, 3.58229529663924854050306231936, 4.54029584454699649646181000533, 5.234962966632513025045586233128, 5.91491092296323377031731714026, 7.309663989561303501406828140299, 8.163591214474489735366776171150, 8.666064254278883966815406218323, 9.57213968124672155163932746178, 10.453491807125811506946397929070, 11.114996805226582415631467847345, 11.65071898634768420648080758332, 12.95716855985823763794140175579, 13.62984278143251065359864884253, 14.47770131421443037918407763671, 15.29259976017474518646318338224, 15.4982426286233248590888264015, 16.69690897073909954136995486054, 17.26077345199723013938041946208, 18.02202231056839816198423579143, 18.985212442076062623294330123717, 19.8540839503352213669645946234, 20.35327885611309995396004791843, 21.23114773722559133352445406913

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