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Properties

Label 1-4033-4033.2512-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.364 + 0.931i$
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 + (0.973 + 0.230i)3-s + (−0.5 + 0.866i)4-s + (−0.0581 + 0.998i)5-s + (−0.286 − 0.957i)6-s + (0.893 − 0.448i)7-s + 8-s + (0.893 + 0.448i)9-s + (0.893 − 0.448i)10-s + (0.893 + 0.448i)11-s + (−0.686 + 0.727i)12-s + (−0.0581 + 0.998i)13-s + (−0.835 − 0.549i)14-s + (−0.286 + 0.957i)15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.973 + 0.230i)3-s + (−0.5 + 0.866i)4-s + (−0.0581 + 0.998i)5-s + (−0.286 − 0.957i)6-s + (0.893 − 0.448i)7-s + 8-s + (0.893 + 0.448i)9-s + (0.893 − 0.448i)10-s + (0.893 + 0.448i)11-s + (−0.686 + 0.727i)12-s + (−0.0581 + 0.998i)13-s + (−0.835 − 0.549i)14-s + (−0.286 + 0.957i)15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.364 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.364 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.364 + 0.931i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2512, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.364 + 0.931i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.718979098 + 1.173676446i\)
\(L(\frac12)\) \(\approx\) \(1.718979098 + 1.173676446i\)
\(L(1)\) \(\approx\) \(1.259199789 + 0.1149712848i\)
\(L(1)\) \(\approx\) \(1.259199789 + 0.1149712848i\)

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 \( 1 + (0.973 + 0.230i)T \)
5 \( 1 + (-0.0581 + 0.998i)T \)
7 \( 1 + (0.893 - 0.448i)T \)
11 \( 1 + (0.893 + 0.448i)T \)
13 \( 1 + (-0.0581 + 0.998i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.173 + 0.984i)T \)
23 \( 1 + (-0.939 - 0.342i)T \)
29 \( 1 + (-0.835 - 0.549i)T \)
31 \( 1 + (0.597 - 0.802i)T \)
41 \( 1 + (0.173 - 0.984i)T \)
43 \( 1 + (0.766 + 0.642i)T \)
47 \( 1 + (-0.286 + 0.957i)T \)
53 \( 1 + (-0.286 - 0.957i)T \)
59 \( 1 + (-0.286 + 0.957i)T \)
61 \( 1 + (0.597 + 0.802i)T \)
67 \( 1 + (-0.686 - 0.727i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (-0.835 + 0.549i)T \)
79 \( 1 + (0.973 - 0.230i)T \)
83 \( 1 + (0.597 + 0.802i)T \)
89 \( 1 + (0.893 - 0.448i)T \)
97 \( 1 + (0.597 + 0.802i)T \)
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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

−18.03852174152217510344577224801, −17.81339056387488444589100481832, −17.058916308740776325246199165270, −16.10239087897771522427138750468, −15.67141915351843507789171689018, −15.021582261297207742118123159375, −14.336148122699299407091122694614, −13.69545121903922358597104395837, −13.19565753452276135470354596043, −12.24421848327351546510271826833, −11.519280449372908997275081447115, −10.574596096552553128086894875224, −9.533186593023440384308945287145, −9.09823682097603282906975785187, −8.60132723168524490235976052787, −7.95027896919266524506769235762, −7.42775874275038505923828299058, −6.513427496182360758082800941592, −5.616231841119903000402065783929, −4.89921157356789250297463867195, −4.313219722734883090026044639879, −3.26785078707400396955404291504, −2.11390425617951615856649927039, −1.38143697689281625873101673004, −0.60769281503337689884909510792, 1.35734682435734235217689580749, 1.976302646512693271363885386548, 2.448842428042581338067980585487, 3.69357583614064518423182853282, 4.027807275228256988523657174734, 4.5120317333383780189205674633, 6.07432010090117868232771487130, 7.00578341446346907612610320408, 7.6826673340638988913073106716, 8.1531538871466019608258445325, 9.02125955586608823654549538773, 9.68136112438007115892160459937, 10.27962483478073459120450989918, 10.9125418883637044917377575180, 11.63008818890005875079074062856, 12.207333649441365254924045151378, 13.27036752525688888567552350532, 13.90053892466393762861110619487, 14.5249276915578498526690666634, 14.80185329362059351319594521605, 15.913781451995130069979565226690, 16.72894595529718260712121827337, 17.467559420042903697063583681833, 18.03459699194007131643420726765, 18.86359589716738228222469890152

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