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Properties

Label 1-3715-3715.1119-r0-0-0
Degree $1$
Conductor $3715$
Sign $-0.818 - 0.573i$
Analytic cond. $17.2523$
Root an. cond. $17.2523$
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  + (0.834 − 0.551i)2-s + (−0.130 + 0.991i)3-s + (0.391 − 0.920i)4-s + (0.437 + 0.899i)6-s + (−0.197 − 0.980i)7-s + (−0.181 − 0.983i)8-s + (−0.965 − 0.259i)9-s + (−0.460 − 0.887i)11-s + (0.861 + 0.508i)12-s + (0.562 − 0.827i)13-s + (−0.705 − 0.708i)14-s + (−0.693 − 0.720i)16-s + (0.987 + 0.160i)17-s + (−0.948 + 0.316i)18-s + (−0.255 + 0.966i)19-s + ⋯
L(s)  = 1  + (0.834 − 0.551i)2-s + (−0.130 + 0.991i)3-s + (0.391 − 0.920i)4-s + (0.437 + 0.899i)6-s + (−0.197 − 0.980i)7-s + (−0.181 − 0.983i)8-s + (−0.965 − 0.259i)9-s + (−0.460 − 0.887i)11-s + (0.861 + 0.508i)12-s + (0.562 − 0.827i)13-s + (−0.705 − 0.708i)14-s + (−0.693 − 0.720i)16-s + (0.987 + 0.160i)17-s + (−0.948 + 0.316i)18-s + (−0.255 + 0.966i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3715 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.818 - 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3715 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.818 - 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3715\)    =    \(5 \cdot 743\)
Sign: $-0.818 - 0.573i$
Analytic conductor: \(17.2523\)
Root analytic conductor: \(17.2523\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3715} (1119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3715,\ (0:\ ),\ -0.818 - 0.573i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5960835933 - 1.889306240i\)
\(L(\frac12)\) \(\approx\) \(0.5960835933 - 1.889306240i\)
\(L(1)\) \(\approx\) \(1.306034782 - 0.6095209097i\)
\(L(1)\) \(\approx\) \(1.306034782 - 0.6095209097i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
743 \( 1 \)
good2 \( 1 + (0.834 - 0.551i)T \)
3 \( 1 + (-0.130 + 0.991i)T \)
7 \( 1 + (-0.197 - 0.980i)T \)
11 \( 1 + (-0.460 - 0.887i)T \)
13 \( 1 + (0.562 - 0.827i)T \)
17 \( 1 + (0.987 + 0.160i)T \)
19 \( 1 + (-0.255 + 0.966i)T \)
23 \( 1 + (-0.406 - 0.913i)T \)
29 \( 1 + (0.881 - 0.471i)T \)
31 \( 1 + (0.998 + 0.0507i)T \)
37 \( 1 + (-0.937 + 0.348i)T \)
41 \( 1 + (0.918 + 0.395i)T \)
43 \( 1 + (-0.931 - 0.364i)T \)
47 \( 1 + (-0.856 + 0.515i)T \)
53 \( 1 + (0.705 - 0.708i)T \)
59 \( 1 + (-0.603 - 0.797i)T \)
61 \( 1 + (0.847 + 0.530i)T \)
67 \( 1 + (0.999 + 0.0254i)T \)
71 \( 1 + (0.988 + 0.151i)T \)
73 \( 1 + (-0.889 + 0.456i)T \)
79 \( 1 + (-0.999 + 0.0423i)T \)
83 \( 1 + (0.139 - 0.990i)T \)
89 \( 1 + (0.230 - 0.973i)T \)
97 \( 1 + (-0.164 - 0.986i)T \)
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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

−18.847235669965833180906549712372, −18.00249734591052420368066262706, −17.67359056370972687747829186232, −16.805256460130351620463897828841, −16.00660477385741011564533305080, −15.48345377271859942957302254813, −14.72337171785080347563469631381, −13.989118148418320700283711269333, −13.437423599256552934670193958386, −12.722552983545618550718013214732, −12.09227619320395481408057136226, −11.75740170712414707548159645366, −10.90458613074469462050747225133, −9.70914579238248322656553461085, −8.79872175002884977892666274231, −8.2176359358262753050587712401, −7.42876296027326222731093799848, −6.75956961157989911694051240025, −6.224922057428179517985771707647, −5.36153047840124535508272874150, −4.91735529007574098690888244529, −3.767629129750768029868260725655, −2.82575471219742136211818090598, −2.27709825194439259266632676128, −1.39373308665719756741342230105, 0.42412844468402237369903986389, 1.271393147299620749965327768017, 2.67944117383212490768100901141, 3.3224386917282846822459348431, 3.82791219926928878274748212763, 4.59261206939613418974163341708, 5.38593680713084389121838607430, 6.0304873519283011343912555513, 6.63864623559625157306693746599, 8.023752088988101160843298654557, 8.42982011384740327232124826233, 9.792138961174424579859538582470, 10.24224980619272230824295205894, 10.561913713855501151500031166950, 11.35609860943181512973248316817, 12.07281156937426046512417130983, 12.899054596184767028648086090961, 13.59866029887353832138301585350, 14.28591354724259248731724084251, 14.67619858666107805351407589733, 15.80901359368702698145875874360, 16.00711762345786849810389928080, 16.77425856861945648659380113316, 17.54135915780529111735727005749, 18.59328947056452389600638903734

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