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Properties

Label 1-4033-4033.631-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.174 - 0.984i$
Analytic cond. $18.7291$
Root an. cond. $18.7291$
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  + (0.939 − 0.342i)2-s + (−0.766 − 0.642i)3-s + (0.766 − 0.642i)4-s + (0.642 − 0.766i)5-s + (−0.939 − 0.342i)6-s + (0.766 + 0.642i)7-s + (0.5 − 0.866i)8-s + (0.173 + 0.984i)9-s + (0.342 − 0.939i)10-s + (−0.342 − 0.939i)11-s − 12-s + (−0.173 + 0.984i)13-s + (0.939 + 0.342i)14-s + (−0.984 + 0.173i)15-s + (0.173 − 0.984i)16-s + (0.766 + 0.642i)17-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)2-s + (−0.766 − 0.642i)3-s + (0.766 − 0.642i)4-s + (0.642 − 0.766i)5-s + (−0.939 − 0.342i)6-s + (0.766 + 0.642i)7-s + (0.5 − 0.866i)8-s + (0.173 + 0.984i)9-s + (0.342 − 0.939i)10-s + (−0.342 − 0.939i)11-s − 12-s + (−0.173 + 0.984i)13-s + (0.939 + 0.342i)14-s + (−0.984 + 0.173i)15-s + (0.173 − 0.984i)16-s + (0.766 + 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.174 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.174 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.174 - 0.984i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (631, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.174 - 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.122742442 - 2.530719627i\)
\(L(\frac12)\) \(\approx\) \(2.122742442 - 2.530719627i\)
\(L(1)\) \(\approx\) \(1.590840283 - 0.9290478839i\)
\(L(1)\) \(\approx\) \(1.590840283 - 0.9290478839i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.939 - 0.342i)T \)
3 \( 1 + (-0.766 - 0.642i)T \)
5 \( 1 + (0.642 - 0.766i)T \)
7 \( 1 + (0.766 + 0.642i)T \)
11 \( 1 + (-0.342 - 0.939i)T \)
13 \( 1 + (-0.173 + 0.984i)T \)
17 \( 1 + (0.766 + 0.642i)T \)
19 \( 1 + (-0.766 + 0.642i)T \)
23 \( 1 + T \)
29 \( 1 + (0.984 - 0.173i)T \)
31 \( 1 + (-0.984 - 0.173i)T \)
41 \( 1 + (-0.342 + 0.939i)T \)
43 \( 1 + (0.866 - 0.5i)T \)
47 \( 1 + (-0.642 + 0.766i)T \)
53 \( 1 + (-0.866 - 0.5i)T \)
59 \( 1 + (-0.173 - 0.984i)T \)
61 \( 1 + (0.642 - 0.766i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (0.939 + 0.342i)T \)
73 \( 1 + (-0.766 - 0.642i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (-0.642 - 0.766i)T \)
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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

−18.23669064758957390083730696099, −17.700878746845258905078549063422, −17.30032231799909996874357999965, −16.6588787426624793625309667083, −15.765573786581754551171886323862, −15.08445668525222425412233902838, −14.72432402944172144094395862637, −14.06840664365230090385105627732, −13.16928382864001779575332769002, −12.598291724147232603491220298025, −11.797036028810535733525803529, −10.98514972426053175734955976603, −10.58777762892895814777368056431, −10.04104578218939738802039565440, −9.01835114473386060070895523765, −7.86235228968751698125057096190, −7.125222616066593257508119453324, −6.75498739920852561964403042931, −5.722564995544676465507244298852, −5.13502393667353548483845128467, −4.7162845169117322744972121690, −3.77019907126720097675722869290, −2.98927595242529070707960373530, −2.20277587728691724308597496443, −1.04300560661216582465304695108, 0.803437741486054678287530262007, 1.692821033398020049881226775045, 2.0252727849943216223253712267, 3.093387703532415255320130740266, 4.27577497354383158876731927119, 4.97006328725805157626178311980, 5.45369946656587156475199620174, 6.141885383535039534852024603815, 6.60226068984217750876737853453, 7.81428910050562240763944323895, 8.37966849992086304619507739399, 9.35659183931087207808213390935, 10.25877463566250487088942549408, 11.06765246342236508107431102927, 11.44027296952952009639239295171, 12.40055959148090239114610134266, 12.60878569167667586131989949682, 13.3488469861362364125410744227, 14.19918714381884624538281825886, 14.467376047099503634824721297739, 15.60799771594407831914474053012, 16.36011169768001311975394609560, 16.83499317897084022395252917642, 17.48281486471801299833970389800, 18.471000593156295992798529289038

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