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Properties

Label 1-4033-4033.64-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.164 - 0.986i$
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  + (−0.5 − 0.866i)2-s + 3-s + (−0.5 + 0.866i)4-s − 5-s + (−0.5 − 0.866i)6-s + 7-s + 8-s + 9-s + (0.5 + 0.866i)10-s + (0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + 13-s + (−0.5 − 0.866i)14-s − 15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + 3-s + (−0.5 + 0.866i)4-s − 5-s + (−0.5 − 0.866i)6-s + 7-s + 8-s + 9-s + (0.5 + 0.866i)10-s + (0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + 13-s + (−0.5 − 0.866i)14-s − 15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.164 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.164 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.164 - 0.986i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.164 - 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.640243988 - 1.389082876i\)
\(L(\frac12)\) \(\approx\) \(1.640243988 - 1.389082876i\)
\(L(1)\) \(\approx\) \(1.124363335 - 0.5229427576i\)
\(L(1)\) \(\approx\) \(1.124363335 - 0.5229427576i\)

Euler product

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

−18.60680408008436868293944961584, −17.9043269128359443802684819336, −17.34997209686516243596993551104, −16.51437925117078567866937258612, −15.546314813099741316763559670301, −15.282535187192423235389278630933, −14.83488274308412058370661634801, −14.10061596660679397294554216149, −13.36615047772760263092522883294, −12.66582358771276612877889782145, −11.636207462756622820507149952, −10.85120057262824126454162062677, −10.32158362842543312099216239963, −9.13268614079832069314234860408, −8.79476300664145650311846224741, −8.1542762336641617814849213480, −7.63489704420680261369225757211, −6.87450713946338067273447956553, −6.310257901117206433267727601557, −4.87366614660243948340114704818, −4.46581689467352630562829692003, −3.874063270506470653661267262617, −2.699819338307330899123087554457, −1.62118606222872962004550134535, −1.03590455445569481645209878032, 0.80392886759461722429936262451, 1.4182936420667954280355529521, 2.41388347308439530804243403183, 3.192080332740726473126994598943, 3.88403863557575849681962387370, 4.32664255156714742997248062573, 5.28223699788560546697505671359, 6.76392988468042682184136477232, 7.39910364551520617906673106669, 8.22509742942355480962934839271, 8.650072104525252645923737571006, 8.93484363339740044086637753972, 10.114934334062245333399664179740, 10.82101167765397324784620227872, 11.43039557000698534242822253723, 11.8956385708579012304115094802, 12.854338300687899925152208676679, 13.489815295717014313507706822026, 14.17611788183219691265057411448, 14.77794726019077372879603380062, 15.652217465530174271242514749209, 16.28637944710376544058635278402, 16.98001313518294276640477784175, 17.98650408158207740093084598129, 18.6112970719171747800239378148

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