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Properties

Label 1-4033-4033.2527-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.709 + 0.704i$
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  + (0.766 + 0.642i)2-s + (0.597 + 0.802i)3-s + (0.173 + 0.984i)4-s + (0.686 − 0.727i)5-s + (−0.0581 + 0.998i)6-s + (0.973 + 0.230i)7-s + (−0.5 + 0.866i)8-s + (−0.286 + 0.957i)9-s + (0.993 − 0.116i)10-s + (0.993 + 0.116i)11-s + (−0.686 + 0.727i)12-s + (0.973 + 0.230i)13-s + (0.597 + 0.802i)14-s + (0.993 + 0.116i)15-s + (−0.939 + 0.342i)16-s + (−0.939 + 0.342i)17-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)2-s + (0.597 + 0.802i)3-s + (0.173 + 0.984i)4-s + (0.686 − 0.727i)5-s + (−0.0581 + 0.998i)6-s + (0.973 + 0.230i)7-s + (−0.5 + 0.866i)8-s + (−0.286 + 0.957i)9-s + (0.993 − 0.116i)10-s + (0.993 + 0.116i)11-s + (−0.686 + 0.727i)12-s + (0.973 + 0.230i)13-s + (0.597 + 0.802i)14-s + (0.993 + 0.116i)15-s + (−0.939 + 0.342i)16-s + (−0.939 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.709 + 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.709 + 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.709 + 0.704i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2527, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.709 + 0.704i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.863376826 + 4.524506872i\)
\(L(\frac12)\) \(\approx\) \(1.863376826 + 4.524506872i\)
\(L(1)\) \(\approx\) \(1.870725192 + 1.675786092i\)
\(L(1)\) \(\approx\) \(1.870725192 + 1.675786092i\)

Euler product

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

−18.428761047776163245468705496156, −17.670386641883015450027222884256, −17.28946852223462754940608831384, −15.780854963754877699734154261154, −15.2138557787270601127135204629, −14.48862768251856302785549200685, −13.86906795248421540281278312232, −13.66033568493883715287355095130, −13.009509184923131487856256871981, −11.87973224533751960140811788004, −11.451356301282748858138004399, −10.96556830510511883887045933109, −9.91883552295931374916242032052, −9.30254948936734973561383220428, −8.56012701632713978838285645427, −7.53954778290913874005444655059, −6.883036377403753048691372391700, −6.09904354355656938013702970971, −5.70307948849275421725415997234, −4.416123986833828901884766175198, −3.84720906508283706126129577025, −2.91956750134526297899784146625, −2.2878978112437537110217922854, −1.54852829680826004525893508688, −0.93372266658328640137910631852, 1.5455979218354066679116074798, 2.00757285033925369366305783021, 3.15992067556091778597494095962, 4.00243176167701873413100225179, 4.493167804243362935507561327503, 5.1652986758898696331127285879, 5.90519466396777096981628945599, 6.551933319356798356465162004166, 7.72216540788876790271189219644, 8.36815935029732841871634291643, 8.88812351503210205529334538991, 9.391061846630487709693432746383, 10.517969666546248153776713872196, 11.232886291654834657711752135, 11.99512574570376724942031068061, 12.70228247769541018334729262251, 13.62503912711640070864010354256, 14.164983731096031414662548423594, 14.3670927490664299856325487623, 15.370232514227694362198388487735, 15.9124238295241083960586589192, 16.51137407692568120613404485593, 17.18433460682854511957888168396, 17.78211540819235704874311808704, 18.497831828804945549846219725544

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