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Properties

Label 1-4033-4033.23-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.633 - 0.773i$
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 + (−0.173 − 0.984i)3-s + (−0.5 − 0.866i)4-s + (0.342 − 0.939i)5-s + (−0.939 − 0.342i)6-s + (−0.939 − 0.342i)7-s − 8-s + (−0.939 + 0.342i)9-s + (−0.642 − 0.766i)10-s + (0.642 − 0.766i)11-s + (−0.766 + 0.642i)12-s + (0.939 + 0.342i)13-s + (−0.766 + 0.642i)14-s + (−0.984 − 0.173i)15-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.173 − 0.984i)3-s + (−0.5 − 0.866i)4-s + (0.342 − 0.939i)5-s + (−0.939 − 0.342i)6-s + (−0.939 − 0.342i)7-s − 8-s + (−0.939 + 0.342i)9-s + (−0.642 − 0.766i)10-s + (0.642 − 0.766i)11-s + (−0.766 + 0.642i)12-s + (0.939 + 0.342i)13-s + (−0.766 + 0.642i)14-s + (−0.984 − 0.173i)15-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.633 - 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.633 - 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.633 - 0.773i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.633 - 0.773i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7209801325 - 0.3412473720i\)
\(L(\frac12)\) \(\approx\) \(0.7209801325 - 0.3412473720i\)
\(L(1)\) \(\approx\) \(0.5958158347 - 0.7807197000i\)
\(L(1)\) \(\approx\) \(0.5958158347 - 0.7807197000i\)

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 \( 1 + (-0.173 - 0.984i)T \)
5 \( 1 + (0.342 - 0.939i)T \)
7 \( 1 + (-0.939 - 0.342i)T \)
11 \( 1 + (0.642 - 0.766i)T \)
13 \( 1 + (0.939 + 0.342i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 - T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.642 + 0.766i)T \)
31 \( 1 + (0.984 + 0.173i)T \)
41 \( 1 + (-0.866 - 0.5i)T \)
43 \( 1 + (-0.866 + 0.5i)T \)
47 \( 1 + (0.342 + 0.939i)T \)
53 \( 1 + (-0.642 - 0.766i)T \)
59 \( 1 + (-0.173 + 0.984i)T \)
61 \( 1 + (0.642 + 0.766i)T \)
67 \( 1 + (0.642 - 0.766i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (-0.766 - 0.642i)T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (0.939 + 0.342i)T \)
89 \( 1 + (0.984 - 0.173i)T \)
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

−18.414488856798091595729183724614, −17.60988129734435883970704541598, −17.14342240773119537126657204650, −16.34941165905838705898575377986, −15.648114799978049180070842589604, −15.29360523045627206589260597719, −14.70310696495643486310370361888, −13.88031196713280571853769059378, −13.42210313936003323974973730825, −12.41156299245522236960827568858, −11.778628410113618495705586342729, −10.98316226970776022845425690182, −9.99328375033816039533574706619, −9.72538355871534785678157859630, −8.76191658714359541500105548769, −8.2422034910674982608420852016, −6.9387848686356240580963056090, −6.439017676555290540421853108456, −6.10741424947801562981918142841, −5.13165860267233565025378423272, −4.30730306639730315865180500755, −3.70836249862902245413474185870, −2.94154376164451611586590573929, −2.28637775921874674197124909336, −0.21295487785641398563972297501, 0.98664169483282442109162290802, 1.43924902001258876472406463609, 2.28484922071054044295314249309, 3.33228155189593348657523918386, 3.92715183153112853404435060380, 4.809730437597162007052665418324, 5.85000810420529454620306392845, 6.23904309682473846669239106909, 6.74386764936214060312673295781, 8.247683172771333349791055888334, 8.66830643561246025516222685886, 9.31535659822073712046252961268, 10.272820228999465799275125303425, 10.92277521379650443965824125690, 11.73695739016514903666535981392, 12.28069717158560205755024231958, 12.973628705444107659360286246386, 13.46740223717600835713121463396, 13.77871064076713375590026949602, 14.64846479141443968032701025872, 15.76571645583308987226794272150, 16.3827092211625275811749414720, 17.25193080505854318731935579539, 17.64414100865714499370344949853, 18.65493527396662797774825719018

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