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Properties

Label 1-3503-3503.204-r0-0-0
Degree $1$
Conductor $3503$
Sign $-0.851 - 0.524i$
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  + (0.995 + 0.0896i)2-s + (0.663 + 0.748i)3-s + (0.983 + 0.178i)4-s + (−0.916 − 0.399i)5-s + (0.593 + 0.804i)6-s + (−0.842 + 0.538i)7-s + (0.963 + 0.266i)8-s + (−0.119 + 0.992i)9-s + (−0.877 − 0.480i)10-s + (0.0299 − 0.999i)11-s + (0.519 + 0.854i)12-s + (−0.894 + 0.447i)13-s + (−0.887 + 0.460i)14-s + (−0.309 − 0.951i)15-s + (0.936 + 0.351i)16-s + (−0.830 + 0.557i)17-s + ⋯
L(s)  = 1  + (0.995 + 0.0896i)2-s + (0.663 + 0.748i)3-s + (0.983 + 0.178i)4-s + (−0.916 − 0.399i)5-s + (0.593 + 0.804i)6-s + (−0.842 + 0.538i)7-s + (0.963 + 0.266i)8-s + (−0.119 + 0.992i)9-s + (−0.877 − 0.480i)10-s + (0.0299 − 0.999i)11-s + (0.519 + 0.854i)12-s + (−0.894 + 0.447i)13-s + (−0.887 + 0.460i)14-s + (−0.309 − 0.951i)15-s + (0.936 + 0.351i)16-s + (−0.830 + 0.557i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3503 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3503 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3503\)    =    \(31 \cdot 113\)
Sign: $-0.851 - 0.524i$
Analytic conductor: \(16.2678\)
Root analytic conductor: \(16.2678\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3503} (204, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3503,\ (0:\ ),\ -0.851 - 0.524i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1934992143 + 0.6833351923i\)
\(L(\frac12)\) \(\approx\) \(-0.1934992143 + 0.6833351923i\)
\(L(1)\) \(\approx\) \(1.353470946 + 0.5549665102i\)
\(L(1)\) \(\approx\) \(1.353470946 + 0.5549665102i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 \)
113 \( 1 \)
good2 \( 1 + (0.995 + 0.0896i)T \)
3 \( 1 + (0.663 + 0.748i)T \)
5 \( 1 + (-0.916 - 0.399i)T \)
7 \( 1 + (-0.842 + 0.538i)T \)
11 \( 1 + (0.0299 - 0.999i)T \)
13 \( 1 + (-0.894 + 0.447i)T \)
17 \( 1 + (-0.830 + 0.557i)T \)
19 \( 1 + (0.995 - 0.0970i)T \)
23 \( 1 + (-0.738 + 0.674i)T \)
29 \( 1 + (-0.640 - 0.767i)T \)
37 \( 1 + (-0.757 - 0.652i)T \)
41 \( 1 + (-0.379 + 0.925i)T \)
43 \( 1 + (0.938 + 0.344i)T \)
47 \( 1 + (-0.999 - 0.0224i)T \)
53 \( 1 + (-0.525 - 0.850i)T \)
59 \( 1 + (0.748 - 0.663i)T \)
61 \( 1 + (-0.433 + 0.900i)T \)
67 \( 1 + (-0.982 - 0.185i)T \)
71 \( 1 + (-0.629 + 0.777i)T \)
73 \( 1 + (0.358 - 0.933i)T \)
79 \( 1 + (-0.727 - 0.685i)T \)
83 \( 1 + (0.575 - 0.817i)T \)
89 \( 1 + (-0.997 - 0.0672i)T \)
97 \( 1 + (-0.963 + 0.266i)T \)
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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

−18.54579255414503193568200491694, −17.77031512164983601804164718363, −16.83743722112919967867251636392, −15.902156481793591304563581788636, −15.476435487400629465805163902003, −14.76220260978320306727701140917, −14.15477035178396308353142601182, −13.54438096258040192857835086873, −12.683730256332033828368671926909, −12.31291320667310820072774457096, −11.753193799109075137546539826981, −10.7494349039716661876450196811, −10.0270858167864748360599865311, −9.2701610688249034029937413350, −8.07873500769552348688041882909, −7.31182173377615250923014229356, −7.085398042792872134390538744243, −6.45017041378563728055470280625, −5.309004510913523085409858678360, −4.37005658403110764190521790270, −3.75250767537092112300917530231, −2.95100017580746350318517234076, −2.48502881267592940766024823617, −1.41063258251869846669770958552, −0.114583475651698871582543203919, 1.712940951015873377622861290385, 2.66883123249243372757113575248, 3.33809465258875544156404758691, 3.87306140560893508911201818820, 4.60800412953999628099513050904, 5.38338676159106167876299805710, 6.08760411769002414063701320739, 7.110140961282997683052633827908, 7.81311046296169246875841823545, 8.52262220420904904377309995568, 9.31230899436241417994662873924, 9.99996380450076634292869365005, 11.04833617880052622797723052605, 11.57174432162998134239401339546, 12.217423222921388064751713760712, 13.13516116951861003088311857828, 13.54558197159411230087785583734, 14.452155168453192397495003030785, 15.04867817261389618653217421002, 15.67279096830834076939604839855, 16.24820468008097167437208456875, 16.46293161540533094945239375041, 17.56025734612424497364761632760, 19.030487376252854789429232367718, 19.37788106491991627437365772104

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