Properties

Label 1-1480-1480.707-r0-0-0
Degree $1$
Conductor $1480$
Sign $0.389 + 0.920i$
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 + (0.642 − 0.766i)13-s + (−0.642 − 0.766i)17-s + (−0.939 + 0.342i)19-s + (−0.173 − 0.984i)21-s + (0.866 + 0.5i)23-s + (−0.866 − 0.5i)27-s + (0.5 + 0.866i)29-s + 31-s + (0.642 − 0.766i)33-s + (0.939 + 0.342i)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 + (0.642 − 0.766i)13-s + (−0.642 − 0.766i)17-s + (−0.939 + 0.342i)19-s + (−0.173 − 0.984i)21-s + (0.866 + 0.5i)23-s + (−0.866 − 0.5i)27-s + (0.5 + 0.866i)29-s + 31-s + (0.642 − 0.766i)33-s + (0.939 + 0.342i)39-s + (0.766 + 0.642i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.389 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.389 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $0.389 + 0.920i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (707, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ 0.389 + 0.920i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.076083187 + 0.7132193756i\)
\(L(\frac12)\) \(\approx\) \(1.076083187 + 0.7132193756i\)
\(L(1)\) \(\approx\) \(0.9682492995 + 0.2787968902i\)
\(L(1)\) \(\approx\) \(0.9682492995 + 0.2787968902i\)

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 \( 1 + (0.642 - 0.766i)T \)
17 \( 1 + (-0.642 - 0.766i)T \)
19 \( 1 + (-0.939 + 0.342i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + T \)
41 \( 1 + (0.766 + 0.642i)T \)
43 \( 1 + iT \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 + (-0.984 + 0.173i)T \)
59 \( 1 + (0.173 + 0.984i)T \)
61 \( 1 + (0.766 + 0.642i)T \)
67 \( 1 + (0.984 + 0.173i)T \)
71 \( 1 + (0.939 - 0.342i)T \)
73 \( 1 - iT \)
79 \( 1 + (-0.173 + 0.984i)T \)
83 \( 1 + (-0.642 - 0.766i)T \)
89 \( 1 + (0.173 + 0.984i)T \)
97 \( 1 + (0.866 + 0.5i)T \)
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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

−20.434674576100937480813941243038, −19.58319136254483680671764572199, −18.98129183777092603912947677844, −18.566302242968906117664421790444, −17.37416876553360107567280294134, −17.13404000016700749593446349203, −15.74979523705047913777281467488, −15.36850223144347758251951288882, −14.345881408712257012381025885695, −13.535900466703238262897431458252, −12.83931302869353385820856871387, −12.470247142873842883923729758837, −11.44434308230917544353762923773, −10.55111807996500791825092729921, −9.59098460085983401526268351576, −8.77496877105062098667542903189, −8.19323439532842175326938657298, −6.964173354098037693431378994252, −6.63742015395049171693723667013, −5.836385281378811856898036217937, −4.53140874528476146745381827925, −3.644128890073470843984602740891, −2.497170096940057256696570974, −2.021792744537163481400760908848, −0.606004767064991741779778439548, 0.85641619965824932750672895763, 2.66446217745441923818062701858, 3.07595333020247708310103892069, 3.97806953290695996417569950980, 4.91088945346105086297298044182, 5.81797100613046702721869421883, 6.548332683665676672778529903031, 7.75322970252957543126405152560, 8.57208662193996696865158208095, 9.20456711326249634833287771943, 10.06763452889828883825685439871, 10.76838175253419549352106693643, 11.29538805501724331126914225752, 12.62035725505126119396262646308, 13.32390381451327808127776072184, 13.913502860182783466666715569621, 14.90749026353232315306820799540, 15.74783071307923173717259650114, 16.04779656646765707322061283851, 16.84349752045931855084562600532, 17.71927941328432835342991267804, 18.735855328408228962511887492733, 19.42721668406760279283003255567, 20.055007127218295844138917425042, 20.91565617263993399414800266448

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