Properties

Label 1-4033-4033.118-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.890 - 0.455i$
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 + (0.396 + 0.918i)3-s + (−0.939 − 0.342i)4-s + (−0.686 − 0.727i)5-s + (0.973 − 0.230i)6-s + (0.973 − 0.230i)7-s + (−0.5 + 0.866i)8-s + (−0.686 + 0.727i)9-s + (−0.835 + 0.549i)10-s + (−0.835 − 0.549i)11-s + (−0.0581 − 0.998i)12-s + (−0.286 + 0.957i)13-s + (−0.0581 − 0.998i)14-s + (0.396 − 0.918i)15-s + (0.766 + 0.642i)16-s + (0.173 − 0.984i)17-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)2-s + (0.396 + 0.918i)3-s + (−0.939 − 0.342i)4-s + (−0.686 − 0.727i)5-s + (0.973 − 0.230i)6-s + (0.973 − 0.230i)7-s + (−0.5 + 0.866i)8-s + (−0.686 + 0.727i)9-s + (−0.835 + 0.549i)10-s + (−0.835 − 0.549i)11-s + (−0.0581 − 0.998i)12-s + (−0.286 + 0.957i)13-s + (−0.0581 − 0.998i)14-s + (0.396 − 0.918i)15-s + (0.766 + 0.642i)16-s + (0.173 − 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.890 - 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.890 - 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.890 - 0.455i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.890 - 0.455i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1837513453 - 0.7631348154i\)
\(L(\frac12)\) \(\approx\) \(0.1837513453 - 0.7631348154i\)
\(L(1)\) \(\approx\) \(0.8593525365 - 0.3330260107i\)
\(L(1)\) \(\approx\) \(0.8593525365 - 0.3330260107i\)

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 \( 1 + (0.396 + 0.918i)T \)
5 \( 1 + (-0.686 - 0.727i)T \)
7 \( 1 + (0.973 - 0.230i)T \)
11 \( 1 + (-0.835 - 0.549i)T \)
13 \( 1 + (-0.286 + 0.957i)T \)
17 \( 1 + (0.173 - 0.984i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.766 + 0.642i)T \)
29 \( 1 + (-0.835 + 0.549i)T \)
31 \( 1 + (-0.993 + 0.116i)T \)
41 \( 1 + (0.766 + 0.642i)T \)
43 \( 1 + (0.766 - 0.642i)T \)
47 \( 1 + (-0.286 - 0.957i)T \)
53 \( 1 + (-0.0581 - 0.998i)T \)
59 \( 1 + (0.396 - 0.918i)T \)
61 \( 1 + (-0.835 + 0.549i)T \)
67 \( 1 + (0.893 - 0.448i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
73 \( 1 + (0.893 - 0.448i)T \)
79 \( 1 + (-0.835 - 0.549i)T \)
83 \( 1 + (-0.686 - 0.727i)T \)
89 \( 1 + (-0.993 - 0.116i)T \)
97 \( 1 + (0.396 + 0.918i)T \)
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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

−18.426596520376918881687668470154, −18.19879331796327087745717802568, −17.39362742037038297579030915505, −17.002661075116574163718719142031, −15.57362523688352986283873571428, −15.33649511768199924371402866417, −14.62799966432115443748455389990, −14.37052183416860151652648799363, −13.2697283135839177738043202931, −12.70603089456523331334874318043, −12.297066194832186351979957813592, −11.120647471450660641094048330464, −10.685977447377547425400304312326, −9.49127602267053905424285357241, −8.619681741997917414166427553328, −8.04003638165059054561381773581, −7.53866257358892404916634369584, −7.136392325987804248744548596767, −6.15496095396502350892690901089, −5.52889091107997373769507223808, −4.62420538039653640502337954070, −3.85523308642816343078258513188, −2.83891817118463030268992099195, −2.28971304039336422909278887942, −0.94180469491077196949181158076, 0.22808214711911979871605507106, 1.474540209928333470673220107355, 2.21793550030832980164987455363, 3.23148828592542515024226439526, 3.870401278980880091367810663225, 4.49409466648898815374561403429, 5.22807069038615519525579854016, 5.496848953527053462318111594919, 7.29653673080147348687222315925, 7.99752633774089161994672440978, 8.64640151646752248236735131980, 9.20350700094420818871481041002, 9.87706812113120190830032057504, 10.851109699987525692319037951846, 11.21969772042947512862522123253, 11.75119719350003708764577180914, 12.67456054437711410273112866126, 13.38009042879128377090733464249, 14.14754478349969206644677557271, 14.61773558561813177308158824876, 15.32930677684092425253993125146, 16.25468108406382517888153749196, 16.69787713795539758906754419632, 17.45945432468282593445548162654, 18.530051260386591637744401314771

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