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Properties

Label 1-3503-3503.1043-r0-0-0
Degree $1$
Conductor $3503$
Sign $0.949 + 0.312i$
Analytic cond. $16.2678$
Root an. cond. $16.2678$
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  + (0.963 + 0.266i)2-s + (−0.904 − 0.427i)3-s + (0.858 + 0.512i)4-s + (−0.982 − 0.185i)5-s + (−0.757 − 0.652i)6-s + (0.791 − 0.611i)7-s + (0.691 + 0.722i)8-s + (0.635 + 0.772i)9-s + (−0.897 − 0.440i)10-s + (−0.817 − 0.575i)11-s + (−0.557 − 0.830i)12-s + (−0.762 − 0.646i)13-s + (0.925 − 0.379i)14-s + (0.809 + 0.587i)15-s + (0.473 + 0.880i)16-s + (−0.748 + 0.663i)17-s + ⋯
L(s)  = 1  + (0.963 + 0.266i)2-s + (−0.904 − 0.427i)3-s + (0.858 + 0.512i)4-s + (−0.982 − 0.185i)5-s + (−0.757 − 0.652i)6-s + (0.791 − 0.611i)7-s + (0.691 + 0.722i)8-s + (0.635 + 0.772i)9-s + (−0.897 − 0.440i)10-s + (−0.817 − 0.575i)11-s + (−0.557 − 0.830i)12-s + (−0.762 − 0.646i)13-s + (0.925 − 0.379i)14-s + (0.809 + 0.587i)15-s + (0.473 + 0.880i)16-s + (−0.748 + 0.663i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3503 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3503 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3503\)    =    \(31 \cdot 113\)
Sign: $0.949 + 0.312i$
Analytic conductor: \(16.2678\)
Root analytic conductor: \(16.2678\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3503} (1043, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3503,\ (0:\ ),\ 0.949 + 0.312i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.721895558 + 0.2763147759i\)
\(L(\frac12)\) \(\approx\) \(1.721895558 + 0.2763147759i\)
\(L(1)\) \(\approx\) \(1.225358256 + 0.02618284706i\)
\(L(1)\) \(\approx\) \(1.225358256 + 0.02618284706i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 \)
113 \( 1 \)
good2 \( 1 + (0.963 + 0.266i)T \)
3 \( 1 + (-0.904 - 0.427i)T \)
5 \( 1 + (-0.982 - 0.185i)T \)
7 \( 1 + (0.791 - 0.611i)T \)
11 \( 1 + (-0.817 - 0.575i)T \)
13 \( 1 + (-0.762 - 0.646i)T \)
17 \( 1 + (-0.748 + 0.663i)T \)
19 \( 1 + (0.727 + 0.685i)T \)
23 \( 1 + (-0.605 - 0.795i)T \)
29 \( 1 + (-0.869 + 0.493i)T \)
37 \( 1 + (0.999 + 0.0373i)T \)
41 \( 1 + (-0.119 + 0.992i)T \)
43 \( 1 + (-0.506 + 0.862i)T \)
47 \( 1 + (0.997 + 0.0672i)T \)
53 \( 1 + (-0.420 + 0.907i)T \)
59 \( 1 + (0.427 - 0.904i)T \)
61 \( 1 + (0.974 - 0.222i)T \)
67 \( 1 + (-0.884 + 0.467i)T \)
71 \( 1 + (0.838 - 0.544i)T \)
73 \( 1 + (-0.0523 + 0.998i)T \)
79 \( 1 + (0.985 + 0.171i)T \)
83 \( 1 + (0.712 - 0.701i)T \)
89 \( 1 + (0.979 + 0.200i)T \)
97 \( 1 + (-0.691 + 0.722i)T \)
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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

−18.66771907351733627178555139344, −18.09229441788101807526570791748, −17.31888748733266586434715842601, −16.32243742980980134020631799380, −15.784820473049867188890473070837, −15.22353603358190705944441444664, −14.87268038285877602255191589788, −13.85881424476605703640295054290, −13.04681883286476463786339567049, −12.08889151160879372225158112898, −11.84942435450515060667286529777, −11.28650590315264290080325752322, −10.68703610513636921770632123473, −9.788799053387382908059922580669, −9.056252791551861626618311096819, −7.643729716633707992636696012399, −7.31983296909537755298891134251, −6.48691990813992066976276509514, −5.3684133221592493966899786175, −5.07270098926966202641519692994, −4.36428486765220615919893645682, −3.72783043327241128941846297888, −2.58610399805441433036929548072, −1.95345381113940912576380974278, −0.56751630128543238684968438656, 0.738529375417456133788794760, 1.78915786400725015271940972500, 2.787370627598740887990178538032, 3.79794351263073728709305177889, 4.51108362620096894552904137525, 5.06450373766909014671567535140, 5.75571625787738876524323789366, 6.59767455668428242843999114686, 7.47811329942513708052140437804, 7.86151564522367872302977072699, 8.352204337852315939445188851000, 10.05318343996607305034538474231, 10.87854576125035571510905476377, 11.155788197989517901550547998641, 11.91853705888517210077067775048, 12.61394963894472970065279267937, 13.05847754571533483431927098894, 13.86696667500272225163482616320, 14.73307728314002191963254232223, 15.25983190502632112340531723103, 16.22010790291624208856514808871, 16.462361085130659770825415074568, 17.247621729300352746344301607731, 17.99088566571762338897538661494, 18.6879609492753732763934069404

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