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Properties

Label 1-3520-3520.1587-r0-0-0
Degree $1$
Conductor $3520$
Sign $0.195 + 0.980i$
Analytic cond. $16.3468$
Root an. cond. $16.3468$
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  + (−0.972 + 0.233i)3-s + (−0.156 − 0.987i)7-s + (0.891 − 0.453i)9-s + (−0.760 − 0.649i)13-s + (−0.309 − 0.951i)17-s + (0.972 − 0.233i)19-s + (0.382 + 0.923i)21-s + (−0.707 + 0.707i)23-s + (−0.760 + 0.649i)27-s + (−0.852 + 0.522i)29-s + (0.309 − 0.951i)31-s + (−0.233 + 0.972i)37-s + (0.891 + 0.453i)39-s + (−0.987 − 0.156i)41-s + (−0.923 + 0.382i)43-s + ⋯
L(s)  = 1  + (−0.972 + 0.233i)3-s + (−0.156 − 0.987i)7-s + (0.891 − 0.453i)9-s + (−0.760 − 0.649i)13-s + (−0.309 − 0.951i)17-s + (0.972 − 0.233i)19-s + (0.382 + 0.923i)21-s + (−0.707 + 0.707i)23-s + (−0.760 + 0.649i)27-s + (−0.852 + 0.522i)29-s + (0.309 − 0.951i)31-s + (−0.233 + 0.972i)37-s + (0.891 + 0.453i)39-s + (−0.987 − 0.156i)41-s + (−0.923 + 0.382i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.195 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.195 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3520\)    =    \(2^{6} \cdot 5 \cdot 11\)
Sign: $0.195 + 0.980i$
Analytic conductor: \(16.3468\)
Root analytic conductor: \(16.3468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3520} (1587, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3520,\ (0:\ ),\ 0.195 + 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3344443176 + 0.2743752933i\)
\(L(\frac12)\) \(\approx\) \(0.3344443176 + 0.2743752933i\)
\(L(1)\) \(\approx\) \(0.6445083217 - 0.04824714886i\)
\(L(1)\) \(\approx\) \(0.6445083217 - 0.04824714886i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good3 \( 1 + (-0.972 + 0.233i)T \)
7 \( 1 + (-0.156 - 0.987i)T \)
13 \( 1 + (-0.760 - 0.649i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (0.972 - 0.233i)T \)
23 \( 1 + (-0.707 + 0.707i)T \)
29 \( 1 + (-0.852 + 0.522i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (-0.233 + 0.972i)T \)
41 \( 1 + (-0.987 - 0.156i)T \)
43 \( 1 + (-0.923 + 0.382i)T \)
47 \( 1 + (-0.809 + 0.587i)T \)
53 \( 1 + (0.0784 + 0.996i)T \)
59 \( 1 + (-0.972 - 0.233i)T \)
61 \( 1 + (0.996 + 0.0784i)T \)
67 \( 1 + (-0.923 - 0.382i)T \)
71 \( 1 + (0.891 + 0.453i)T \)
73 \( 1 + (0.987 - 0.156i)T \)
79 \( 1 + (-0.951 - 0.309i)T \)
83 \( 1 + (0.996 + 0.0784i)T \)
89 \( 1 + (-0.707 + 0.707i)T \)
97 \( 1 + (-0.951 - 0.309i)T \)
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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

−18.36355103712573930184391121054, −18.064472911483738204634055789029, −17.17477935574776966636413204351, −16.574208107598414691127910432191, −15.961304160199501235597308323540, −15.23897847285790467564164333000, −14.539084401376860386217797853514, −13.644915805341872193396021677394, −12.82555907366312814530929028556, −12.20235465079222008831185655577, −11.787504577721221934661450844036, −11.06102900391512275684119272071, −10.12415869368976477861840376999, −9.66063576648876128846328757865, −8.67353945257397489717374858484, −7.967610788110806617711421929303, −6.97756551667218382697705720608, −6.47072844529891708633865788323, −5.63002310424911504115981630305, −5.11980326932508940266421368674, −4.25981154513679949996858847371, −3.31562386664286513673473668541, −2.100300650225623159704249899903, −1.687417586422594562821693160312, −0.19113211904047198700750363142, 0.76605131663615237568974442642, 1.67962651029035407551292930529, 2.99927393109193418935187153790, 3.698295817449439528881199076153, 4.68279519076515294964195012659, 5.121993255684358529417626035574, 5.98046390959314169942423131489, 6.86886859095483642055957178403, 7.37509752329104465372555998860, 8.06982231756066105715905267604, 9.54488792277492393435396127922, 9.70695352534886803837711523974, 10.513106691674856354510180243787, 11.2969704517732047563607671264, 11.77549374152036862898745505085, 12.57669568657309910191721796111, 13.40672982867985882118190584693, 13.84784624972555369719166572105, 14.94472746222178212807752897263, 15.548547151552478426219252166828, 16.27888434591770039993022381446, 16.85617879889667207122837933949, 17.43351039720272496996923812978, 18.077993383783323224167760439550, 18.67619865335399579892828341079

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