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Properties

Label 1-4033-4033.3798-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.328 + 0.944i$
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  + (−0.766 + 0.642i)2-s + (−0.173 − 0.984i)3-s + (0.173 − 0.984i)4-s + (0.984 − 0.173i)5-s + (0.766 + 0.642i)6-s + (0.173 + 0.984i)7-s + (0.5 + 0.866i)8-s + (−0.939 + 0.342i)9-s + (−0.642 + 0.766i)10-s + (0.642 + 0.766i)11-s − 12-s + (0.939 + 0.342i)13-s + (−0.766 − 0.642i)14-s + (−0.342 − 0.939i)15-s + (−0.939 − 0.342i)16-s + (0.173 + 0.984i)17-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)2-s + (−0.173 − 0.984i)3-s + (0.173 − 0.984i)4-s + (0.984 − 0.173i)5-s + (0.766 + 0.642i)6-s + (0.173 + 0.984i)7-s + (0.5 + 0.866i)8-s + (−0.939 + 0.342i)9-s + (−0.642 + 0.766i)10-s + (0.642 + 0.766i)11-s − 12-s + (0.939 + 0.342i)13-s + (−0.766 − 0.642i)14-s + (−0.342 − 0.939i)15-s + (−0.939 − 0.342i)16-s + (0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.328 + 0.944i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (3798, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.328 + 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.266604437 + 0.9008685368i\)
\(L(\frac12)\) \(\approx\) \(1.266604437 + 0.9008685368i\)
\(L(1)\) \(\approx\) \(0.9310634599 + 0.2159788932i\)
\(L(1)\) \(\approx\) \(0.9310634599 + 0.2159788932i\)

Euler product

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

−18.06794103425052642380507804258, −17.72676612114501224668194756502, −16.89557808889149919726207466833, −16.63035316002946002651475039623, −15.92201271588632338922088941459, −14.967303708910059724596776375299, −14.16720029243061648742549852638, −13.34699627151616781913121544081, −13.15935647526106486878812577408, −11.502239504031961192823086184352, −11.40856721333351782063673325176, −10.70054969420111750138859448008, −10.013784320826348413182309877378, −9.50167997576471599178593931903, −8.84372683146438759596455948329, −8.21410978457642977912705659387, −7.09136675954362462362745970273, −6.46187397608544900157802398552, −5.586843868755396596053232190995, −4.62627085622956087550247076319, −3.92385870368833259039825544875, −3.11108110196719642025735595299, −2.56123741217782620401883145773, −1.20988373543832918133792788457, −0.667249963706962782808112512694, 1.16762223581536059729298152573, 1.695752310685773489325226424376, 2.11991158818567172825628779109, 3.3612788071254736068665145170, 4.85209959863864702223255187923, 5.48402181968901559158181086267, 6.14494336620317664541349999314, 6.59309818668885649897147629314, 7.29047429240323827434832488227, 8.433108712586878078496815106196, 8.65628713556339289085655769127, 9.32540438421260557053151188700, 10.265242275843361128230689834801, 10.89778856236179255574558127110, 11.78917327569077154854994278685, 12.50298053057003502337543178991, 13.10317821066124062483852070502, 14.01448904067446012605759764004, 14.59984215679331156609529705118, 15.02601352064960136243072421254, 16.212214936311616931861180546691, 16.69449418192209931675505568717, 17.40830302544754861180970450380, 18.04034894689090394653294115024, 18.24145116698108042160663152698

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