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Properties

Label 1-3520-3520.139-r0-0-0
Degree $1$
Conductor $3520$
Sign $-0.184 + 0.982i$
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 + (0.987 + 0.156i)7-s + (0.891 + 0.453i)9-s + (0.760 − 0.649i)13-s + (−0.951 − 0.309i)17-s + (0.233 − 0.972i)19-s + (−0.923 − 0.382i)21-s + (−0.707 + 0.707i)23-s + (−0.760 − 0.649i)27-s + (−0.522 + 0.852i)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 + (0.987 + 0.156i)7-s + (0.891 + 0.453i)9-s + (0.760 − 0.649i)13-s + (−0.951 − 0.309i)17-s + (0.233 − 0.972i)19-s + (−0.923 − 0.382i)21-s + (−0.707 + 0.707i)23-s + (−0.760 − 0.649i)27-s + (−0.522 + 0.852i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.184 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.184 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3520\)    =    \(2^{6} \cdot 5 \cdot 11\)
Sign: $-0.184 + 0.982i$
Analytic conductor: \(16.3468\)
Root analytic conductor: \(16.3468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3520} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3520,\ (0:\ ),\ -0.184 + 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4538675922 + 0.5470845294i\)
\(L(\frac12)\) \(\approx\) \(0.4538675922 + 0.5470845294i\)
\(L(1)\) \(\approx\) \(0.7865628045 + 0.01771321465i\)
\(L(1)\) \(\approx\) \(0.7865628045 + 0.01771321465i\)

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 \( 1 + (0.987 + 0.156i)T \)
13 \( 1 + (0.760 - 0.649i)T \)
17 \( 1 + (-0.951 - 0.309i)T \)
19 \( 1 + (0.233 - 0.972i)T \)
23 \( 1 + (-0.707 + 0.707i)T \)
29 \( 1 + (-0.522 + 0.852i)T \)
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 \( 1 + (-0.587 + 0.809i)T \)
53 \( 1 + (-0.0784 + 0.996i)T \)
59 \( 1 + (0.233 + 0.972i)T \)
61 \( 1 + (0.0784 + 0.996i)T \)
67 \( 1 + (0.923 - 0.382i)T \)
71 \( 1 + (-0.891 + 0.453i)T \)
73 \( 1 + (-0.156 + 0.987i)T \)
79 \( 1 + (-0.951 + 0.309i)T \)
83 \( 1 + (-0.996 + 0.0784i)T \)
89 \( 1 + (-0.707 - 0.707i)T \)
97 \( 1 + (0.309 + 0.951i)T \)
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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

−18.48929870718584052335940790075, −17.83337825592346067902966733916, −17.09162716414858684285148834800, −16.65409436553728171156271502002, −15.85544083670268276060713892682, −15.18886746245517429536444697273, −14.50268922711408350158422639884, −13.61585887643502511798145211637, −13.04082864892832519592750913700, −11.97869237645009200704918999121, −11.55813351885644620901958734146, −11.03213049355804206239696131078, −10.194634600365528891837504353315, −9.65838481120130549085444282579, −8.47008285743220808580479744926, −8.09384889398074844420914488211, −6.95571820065452508598919674804, −6.36303686868302620565655209746, −5.670600561578353115509571541266, −4.78589515517525761008585510998, −4.23028339717173937612831818701, −3.53768239268391157923510703249, −1.96596295293302788448138940641, −1.5624368835893628421459706897, −0.25073512880518826994910744482, 1.09173390856699290052111798442, 1.710983862068281082379544704691, 2.75221236875858719625178259322, 3.8840321011491256194070289507, 4.71117983246083379713387003555, 5.32469639329306281755270322891, 5.93118859651687394704748634679, 6.91877716155950280970027384989, 7.396332671771852237739628749430, 8.3753703770618685319188655766, 8.95121952255147684834165754342, 10.03829273332723071031306555211, 10.76915431978974868084361069187, 11.293949723217456967141482974719, 11.79291676529377306230315685057, 12.655837892418081827392595199891, 13.3620545318969687556028570591, 13.93108529773669414534819274009, 14.927924185871853204996811212241, 15.72310230796047509801054002323, 16.011963453402284679183001462148, 17.12904538869981879373316206015, 17.61196269327255907217806082188, 18.15458983326732334456109275998, 18.5482965390145619247085015307

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