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Properties

Label 1-4033-4033.2156-r1-0-0
Degree $1$
Conductor $4033$
Sign $-0.633 - 0.773i$
Analytic cond. $433.406$
Root an. cond. $433.406$
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  + (0.642 − 0.766i)2-s + (−0.286 − 0.957i)3-s + (−0.173 − 0.984i)4-s + (−0.0581 + 0.998i)5-s + (−0.918 − 0.396i)6-s + (0.893 − 0.448i)7-s + (−0.866 − 0.5i)8-s + (−0.835 + 0.549i)9-s + (0.727 + 0.686i)10-s + (0.727 − 0.686i)11-s + (−0.893 + 0.448i)12-s + (0.448 + 0.893i)13-s + (0.230 − 0.973i)14-s + (0.973 − 0.230i)15-s + (−0.939 + 0.342i)16-s + (0.342 + 0.939i)17-s + ⋯
L(s)  = 1  + (0.642 − 0.766i)2-s + (−0.286 − 0.957i)3-s + (−0.173 − 0.984i)4-s + (−0.0581 + 0.998i)5-s + (−0.918 − 0.396i)6-s + (0.893 − 0.448i)7-s + (−0.866 − 0.5i)8-s + (−0.835 + 0.549i)9-s + (0.727 + 0.686i)10-s + (0.727 − 0.686i)11-s + (−0.893 + 0.448i)12-s + (0.448 + 0.893i)13-s + (0.230 − 0.973i)14-s + (0.973 − 0.230i)15-s + (−0.939 + 0.342i)16-s + (0.342 + 0.939i)17-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.633 - 0.773i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2156, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ -0.633 - 0.773i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.414932712 - 2.988302282i\)
\(L(\frac12)\) \(\approx\) \(1.414932712 - 2.988302282i\)
\(L(1)\) \(\approx\) \(1.134404402 - 0.9409945196i\)
\(L(1)\) \(\approx\) \(1.134404402 - 0.9409945196i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.642 - 0.766i)T \)
3 \( 1 + (-0.286 - 0.957i)T \)
5 \( 1 + (-0.0581 + 0.998i)T \)
7 \( 1 + (0.893 - 0.448i)T \)
11 \( 1 + (0.727 - 0.686i)T \)
13 \( 1 + (0.448 + 0.893i)T \)
17 \( 1 + (0.342 + 0.939i)T \)
19 \( 1 + (-0.342 - 0.939i)T \)
23 \( 1 + (-0.642 - 0.766i)T \)
29 \( 1 + (0.286 + 0.957i)T \)
31 \( 1 + (0.835 - 0.549i)T \)
41 \( 1 - iT \)
43 \( 1 + (0.939 - 0.342i)T \)
47 \( 1 + (-0.802 - 0.597i)T \)
53 \( 1 + (-0.549 + 0.835i)T \)
59 \( 1 + (0.230 + 0.973i)T \)
61 \( 1 + (0.993 - 0.116i)T \)
67 \( 1 + (0.998 - 0.0581i)T \)
71 \( 1 + (-0.766 + 0.642i)T \)
73 \( 1 + (-0.686 - 0.727i)T \)
79 \( 1 + (-0.448 + 0.893i)T \)
83 \( 1 + (0.286 + 0.957i)T \)
89 \( 1 + (-0.993 + 0.116i)T \)
97 \( 1 + (-0.0581 + 0.998i)T \)
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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

−17.97791737208967371771228939121, −17.64988940340112798087595268902, −17.1072048593930434389525116418, −16.31938225759135548667088241806, −15.755787852157492891285180321631, −15.34906862669513443797189391611, −14.41094107125799171530800974321, −14.16580059942299799823054432537, −13.04624599768914552211052050807, −12.374803544276268656555281726760, −11.67329278906057384175775162291, −11.40422969758009109990841054115, −9.979862564223312569518582789914, −9.55009540108770902067018862938, −8.56819749797407712083083327047, −8.212335321704812742082547522, −7.46624577453063506390524061285, −6.13585186210912204475176115541, −5.798686747514913916507680632006, −4.899825762462082785425743919884, −4.60442744902309230651934863439, −3.83565770195646224115045335569, −3.0189243906025472704754259156, −1.82396964522445790855232619791, −0.69438419158557661052737581446, 0.5482809170331169542050579454, 1.3292141560528515426140122922, 2.04621334985637605378859767813, 2.73033791327063959909335986809, 3.76747739110372805613654343040, 4.27826102491321135142440331446, 5.34375103016088041088162733236, 6.19754724864620070951820193219, 6.58503743700489124093859567478, 7.31211143042167574722006813389, 8.37102477670437087146953063000, 8.89886026174658057424688564928, 10.17946138020342523863350039171, 10.79129059153495274785819690967, 11.288255984726898740918714965028, 11.75348970847824740856260139672, 12.47167555673960520889510979850, 13.3860429275433614702591811691, 14.00302183012379485222692268489, 14.301720797697644247297067800585, 14.9165094244416933647345310422, 15.93013963938014833152663374240, 16.95459697950880176304339384872, 17.55446094633695349288357204631, 18.28037218370511429439684672049

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