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Properties

Label 1-4033-4033.3132-r1-0-0
Degree $1$
Conductor $4033$
Sign $0.710 - 0.703i$
Analytic cond. $433.406$
Root an. cond. $433.406$
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  + i·2-s + (−0.396 + 0.918i)3-s − 4-s + (−0.957 + 0.286i)5-s + (−0.918 − 0.396i)6-s + (−0.686 + 0.727i)7-s − i·8-s + (−0.686 − 0.727i)9-s + (−0.286 − 0.957i)10-s + (0.286 − 0.957i)11-s + (0.396 − 0.918i)12-s + (−0.957 + 0.286i)13-s + (−0.727 − 0.686i)14-s + (0.116 − 0.993i)15-s + 16-s + (−0.866 + 0.5i)17-s + ⋯
L(s)  = 1  + i·2-s + (−0.396 + 0.918i)3-s − 4-s + (−0.957 + 0.286i)5-s + (−0.918 − 0.396i)6-s + (−0.686 + 0.727i)7-s − i·8-s + (−0.686 − 0.727i)9-s + (−0.286 − 0.957i)10-s + (0.286 − 0.957i)11-s + (0.396 − 0.918i)12-s + (−0.957 + 0.286i)13-s + (−0.727 − 0.686i)14-s + (0.116 − 0.993i)15-s + 16-s + (−0.866 + 0.5i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.710 - 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.710 - 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.710 - 0.703i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (3132, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ 0.710 - 0.703i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4749716031 + 0.1952959250i\)
\(L(\frac12)\) \(\approx\) \(-0.4749716031 + 0.1952959250i\)
\(L(1)\) \(\approx\) \(0.2856575807 + 0.4548965458i\)
\(L(1)\) \(\approx\) \(0.2856575807 + 0.4548965458i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + iT \)
3 \( 1 + (-0.396 + 0.918i)T \)
5 \( 1 + (-0.957 + 0.286i)T \)
7 \( 1 + (-0.686 + 0.727i)T \)
11 \( 1 + (0.286 - 0.957i)T \)
13 \( 1 + (-0.957 + 0.286i)T \)
17 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (-0.342 + 0.939i)T \)
23 \( 1 + (0.984 - 0.173i)T \)
29 \( 1 + (-0.727 - 0.686i)T \)
31 \( 1 + (-0.549 - 0.835i)T \)
41 \( 1 + (-0.173 + 0.984i)T \)
43 \( 1 + (0.342 + 0.939i)T \)
47 \( 1 + (0.396 + 0.918i)T \)
53 \( 1 + (0.597 + 0.802i)T \)
59 \( 1 + (-0.116 + 0.993i)T \)
61 \( 1 + (-0.998 - 0.0581i)T \)
67 \( 1 + (-0.396 - 0.918i)T \)
71 \( 1 + T \)
73 \( 1 + (0.686 + 0.727i)T \)
79 \( 1 + (-0.116 + 0.993i)T \)
83 \( 1 + (0.893 - 0.448i)T \)
89 \( 1 + (-0.230 + 0.973i)T \)
97 \( 1 + (-0.549 + 0.835i)T \)
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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

−17.76251150147866659438724048543, −17.15349946029933244645617835235, −16.72944984869811227602417849953, −15.626284424771747620257125094469, −14.87371964246119695334697256105, −14.050438677007523866063606214591, −13.17680306130582812259087651853, −12.84077953698225448786827640143, −12.232234157132498476297616359214, −11.66163997692569541690933459300, −10.8580736314243893522463132470, −10.43915826184411346601563357858, −9.23558606838893674774418067644, −8.91412758976644583532465819171, −7.79723260708808801182011032670, −7.02132616264817134127140870364, −6.91526219486160995153666575824, −5.27833858407816597341302433019, −4.871277286613292659986485193330, −4.025501559261225318496962759058, −3.19186849952234004810616743047, −2.38091382400890402713145880232, −1.55003323191859204328388604803, −0.40047067396415197443215494412, −0.25647031106497862152359227702, 0.78833218434415321041682915079, 2.60140158374700155673448419779, 3.42598164108449144956329434761, 4.08695374206219937382422741307, 4.66897849107785558400169343677, 5.65000551094355177032935879385, 6.178411702546103811255954846876, 6.792742029705930606328328541234, 7.78973642756144223368237104210, 8.455879762193981381446180560008, 9.22637895723942432794760377327, 9.58692310490588155936602798772, 10.6670600962152019155379026620, 11.22936274192609064165556620222, 12.127106146160594161590326762769, 12.650460926454059989906007602107, 13.59830414555180419974934328959, 14.611945686842324008078146789707, 15.02364038386763897953464151956, 15.40680731751746299349368937501, 16.253891865319734428929499353484, 16.67476843961044249847515292981, 17.11790887383444081029831171154, 18.20888094784989248366814697636, 18.841867642399524140639221947762

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