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Properties

Label 1-3520-3520.1483-r0-0-0
Degree $1$
Conductor $3520$
Sign $0.843 - 0.536i$
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  + (−0.996 + 0.0784i)3-s + (−0.891 − 0.453i)7-s + (0.987 − 0.156i)9-s + (−0.972 − 0.233i)13-s + (0.809 − 0.587i)17-s + (0.996 − 0.0784i)19-s + (0.923 + 0.382i)21-s + (0.707 + 0.707i)23-s + (−0.972 + 0.233i)27-s + (0.649 + 0.760i)29-s + (−0.809 − 0.587i)31-s + (0.0784 − 0.996i)37-s + (0.987 + 0.156i)39-s + (0.453 + 0.891i)41-s + (0.382 − 0.923i)43-s + ⋯
L(s)  = 1  + (−0.996 + 0.0784i)3-s + (−0.891 − 0.453i)7-s + (0.987 − 0.156i)9-s + (−0.972 − 0.233i)13-s + (0.809 − 0.587i)17-s + (0.996 − 0.0784i)19-s + (0.923 + 0.382i)21-s + (0.707 + 0.707i)23-s + (−0.972 + 0.233i)27-s + (0.649 + 0.760i)29-s + (−0.809 − 0.587i)31-s + (0.0784 − 0.996i)37-s + (0.987 + 0.156i)39-s + (0.453 + 0.891i)41-s + (0.382 − 0.923i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.843 - 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.843 - 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3520\)    =    \(2^{6} \cdot 5 \cdot 11\)
Sign: $0.843 - 0.536i$
Analytic conductor: \(16.3468\)
Root analytic conductor: \(16.3468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3520} (1483, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3520,\ (0:\ ),\ 0.843 - 0.536i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8928045902 - 0.2599806527i\)
\(L(\frac12)\) \(\approx\) \(0.8928045902 - 0.2599806527i\)
\(L(1)\) \(\approx\) \(0.7256039692 - 0.05574269090i\)
\(L(1)\) \(\approx\) \(0.7256039692 - 0.05574269090i\)

Euler product

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

−18.830883567613280683207698518558, −18.129621326418933622948304882312, −17.26808903981015568861761035049, −16.82265220634527184166174491634, −16.105576927751590562439649659513, −15.581390614786970109210621787260, −14.73016894551684971396448249960, −13.94550220994833832697709299520, −13.003763755910846373292401354647, −12.32555971533510516502383519697, −12.13591841079035117186115594609, −11.12766058591973614262749762010, −10.43418310113177342614014798790, −9.676054798169002867554533185854, −9.270279568888485330602261492402, −8.021690089991178318691981030431, −7.34373897707083380755007039920, −6.52671745359031626401736768751, −6.01676467231719671409133592863, −5.17204748845008964670315530658, −4.595178931096654326096700300735, −3.49830118460205528105134809722, −2.74594860346440747220911899760, −1.67069170882399278776437544008, −0.65789263762029289327466282230, 0.53357991041524832883409455173, 1.32357312995301683616930521294, 2.667972195958988496811096708244, 3.43097767611099761422508831997, 4.27196175281211579178186650955, 5.266792630587785676084477538754, 5.57415369698568357652573350250, 6.62024880370534479840750492560, 7.301315127716013783079515684531, 7.64735387537825672519219825633, 9.20778543070605722448596968426, 9.59531665896724342547340000028, 10.30256919518195566205074055502, 10.9646225551752239860361752301, 11.79493262198612429390695626469, 12.385682065714538974138502978066, 12.96040411273092403769346427469, 13.75968437098015945341885844378, 14.541890434343899307437387231231, 15.47473922858745867276184685479, 16.05248527373494969345047870446, 16.68140655126870915357874029091, 17.15330721368771208491593854101, 17.95730950673440076522152045299, 18.571178811259564815235549530182

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