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Properties

Label 1-3520-3520.1203-r0-0-0
Degree $1$
Conductor $3520$
Sign $0.000343 + 0.999i$
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  + (0.522 + 0.852i)3-s + (0.987 + 0.156i)7-s + (−0.453 + 0.891i)9-s + (−0.996 − 0.0784i)13-s + (0.309 − 0.951i)17-s + (0.522 + 0.852i)19-s + (0.382 + 0.923i)21-s + (0.707 − 0.707i)23-s + (−0.996 + 0.0784i)27-s + (0.233 + 0.972i)29-s + (0.309 + 0.951i)31-s + (−0.852 − 0.522i)37-s + (−0.453 − 0.891i)39-s + (−0.156 − 0.987i)41-s + (0.923 − 0.382i)43-s + ⋯
L(s)  = 1  + (0.522 + 0.852i)3-s + (0.987 + 0.156i)7-s + (−0.453 + 0.891i)9-s + (−0.996 − 0.0784i)13-s + (0.309 − 0.951i)17-s + (0.522 + 0.852i)19-s + (0.382 + 0.923i)21-s + (0.707 − 0.707i)23-s + (−0.996 + 0.0784i)27-s + (0.233 + 0.972i)29-s + (0.309 + 0.951i)31-s + (−0.852 − 0.522i)37-s + (−0.453 − 0.891i)39-s + (−0.156 − 0.987i)41-s + (0.923 − 0.382i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.000343 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.000343 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3520\)    =    \(2^{6} \cdot 5 \cdot 11\)
Sign: $0.000343 + 0.999i$
Analytic conductor: \(16.3468\)
Root analytic conductor: \(16.3468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3520} (1203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3520,\ (0:\ ),\ 0.000343 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.611392802 + 1.611946125i\)
\(L(\frac12)\) \(\approx\) \(1.611392802 + 1.611946125i\)
\(L(1)\) \(\approx\) \(1.286220630 + 0.5152577364i\)
\(L(1)\) \(\approx\) \(1.286220630 + 0.5152577364i\)

Euler product

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

−18.62055441122283539235983648522, −17.82268388947527862716775960450, −17.27890196393663303423671192787, −16.88751410990412882016415334541, −15.464066285626399527029022882147, −15.06016663362752426289950955457, −14.405240966752670897502724344322, −13.67973352635482434119132671911, −13.17510954696053266865749633667, −12.27329740682496617765439945026, −11.71027770139963690784650837082, −11.08677282567968430252407121902, −10.05662725640972687334830258828, −9.336498271949762549908304989515, −8.54484671106622581295048789508, −7.82780229110678469056027247104, −7.402671591273994190783167386224, −6.5965704207588835589062997097, −5.69529055616957413700626789892, −4.89623585151501445592967522003, −4.05776537946205849252692555217, −3.07058875804649863435441753950, −2.29602829533050212703258475937, −1.56659571238829059215322508324, −0.67688149906286064005923975170, 1.06157617374264811156414575880, 2.17201040329983627107961287267, 2.82148039187205868014148131365, 3.66542602486233267696898479531, 4.62886622975606979071362917531, 5.08688117386439316318428425023, 5.71246413516745541092142529782, 7.16487964923887410968637218700, 7.54397914790775243963872681547, 8.51541807263932814344492539310, 9.00344406071380603544943528763, 9.772024237891309237754913168531, 10.59682533023780574810447253792, 10.96441748263267028341978629743, 12.16567413168541656456289794556, 12.33055735740856975890059506496, 13.90627501601007630314838399864, 14.041309490457666539710422927112, 14.74554833990951822526819376719, 15.404331025013005285401950646656, 16.10925441554136564047122378260, 16.793442185681606914699719016441, 17.438985514303611253649064001947, 18.24438220029728745951955879491, 18.95609738122334868119249717728

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