Properties

Label 1-4033-4033.6-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.470 + 0.882i$
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  + (−0.173 + 0.984i)2-s + (0.993 − 0.116i)3-s + (−0.939 − 0.342i)4-s + (0.957 − 0.286i)5-s + (−0.0581 + 0.998i)6-s + (−0.686 + 0.727i)7-s + (0.5 − 0.866i)8-s + (0.973 − 0.230i)9-s + (0.116 + 0.993i)10-s + (−0.116 + 0.993i)11-s + (−0.973 − 0.230i)12-s + (0.686 − 0.727i)13-s + (−0.597 − 0.802i)14-s + (0.918 − 0.396i)15-s + (0.766 + 0.642i)16-s + (0.766 + 0.642i)17-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)2-s + (0.993 − 0.116i)3-s + (−0.939 − 0.342i)4-s + (0.957 − 0.286i)5-s + (−0.0581 + 0.998i)6-s + (−0.686 + 0.727i)7-s + (0.5 − 0.866i)8-s + (0.973 − 0.230i)9-s + (0.116 + 0.993i)10-s + (−0.116 + 0.993i)11-s + (−0.973 − 0.230i)12-s + (0.686 − 0.727i)13-s + (−0.597 − 0.802i)14-s + (0.918 − 0.396i)15-s + (0.766 + 0.642i)16-s + (0.766 + 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.470 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.470 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.470 + 0.882i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.470 + 0.882i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.447905012 + 1.469301660i\)
\(L(\frac12)\) \(\approx\) \(2.447905012 + 1.469301660i\)
\(L(1)\) \(\approx\) \(1.433672418 + 0.6403797249i\)
\(L(1)\) \(\approx\) \(1.433672418 + 0.6403797249i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.173 + 0.984i)T \)
3 \( 1 + (0.993 - 0.116i)T \)
5 \( 1 + (0.957 - 0.286i)T \)
7 \( 1 + (-0.686 + 0.727i)T \)
11 \( 1 + (-0.116 + 0.993i)T \)
13 \( 1 + (0.686 - 0.727i)T \)
17 \( 1 + (0.766 + 0.642i)T \)
19 \( 1 + (0.939 - 0.342i)T \)
23 \( 1 + (-0.939 - 0.342i)T \)
29 \( 1 + (0.918 + 0.396i)T \)
31 \( 1 + (0.727 - 0.686i)T \)
41 \( 1 + (0.866 - 0.5i)T \)
43 \( 1 + (-0.984 - 0.173i)T \)
47 \( 1 + (0.549 + 0.835i)T \)
53 \( 1 + (-0.957 + 0.286i)T \)
59 \( 1 + (-0.396 - 0.918i)T \)
61 \( 1 + (0.549 - 0.835i)T \)
67 \( 1 + (-0.727 + 0.686i)T \)
71 \( 1 + (-0.173 - 0.984i)T \)
73 \( 1 + (0.993 + 0.116i)T \)
79 \( 1 + (-0.973 + 0.230i)T \)
83 \( 1 + (-0.597 - 0.802i)T \)
89 \( 1 + (0.998 + 0.0581i)T \)
97 \( 1 + (-0.230 + 0.973i)T \)
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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

−18.28683794611066450630977400764, −18.20989987729080230429181321479, −16.99985279292127950301095290569, −16.3416104410911181217916097338, −15.8331213944941660571390779657, −14.42876283303750559113963057473, −13.91645897849575532680417266595, −13.729476377144840586105003003760, −13.141192935589520844647270743905, −12.18771529533507707207794653077, −11.408498004048378143879768606948, −10.488534692167628234370581210740, −9.995794066425735139018639635713, −9.56653894868270924024694234191, −8.82634054909140283854050986533, −8.12521664165742746814340509165, −7.32314243277635109668798209538, −6.40610618840367424430110874550, −5.55652234075513670406511850311, −4.536109569823573803993486879109, −3.69301300790917149368050632371, −3.1302601017763027795904090281, −2.62306596135461868560540888410, −1.5029981449803441769033708005, −0.99565347114942213912369496049, 0.94355252834409354173525419633, 1.796507828636842687789872194938, 2.730317998243699529905692959874, 3.51243500694599122730373503023, 4.50570469415430223308598843144, 5.283666221378953216440606254650, 6.08834164231436140448253862810, 6.53478680005022409416832694590, 7.53751686839564443103421827351, 8.15219809521768474698454218788, 8.79425840585925601043596856070, 9.53659431647865691736533015989, 9.89465821066464562654789720976, 10.49119950519221465885845739700, 12.239915568026997027612661776142, 12.66277157082387369027093893984, 13.27723299817757656395443914064, 13.95258555525479915228515952535, 14.492648797125103634394212067115, 15.283897785453956291581890397764, 15.796055803249138230583854197517, 16.30325360849170244793873353890, 17.41887018745192953789860043629, 17.82600676043241120926357424487, 18.517920903901074292950256468313

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