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Properties

Label 1-4033-4033.102-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.389 + 0.921i$
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  + (−0.939 − 0.342i)2-s + (0.396 − 0.918i)3-s + (0.766 + 0.642i)4-s + (0.686 − 0.727i)5-s + (−0.686 + 0.727i)6-s + (0.973 + 0.230i)7-s + (−0.5 − 0.866i)8-s + (−0.686 − 0.727i)9-s + (−0.893 + 0.448i)10-s + (−0.893 − 0.448i)11-s + (0.893 − 0.448i)12-s + (−0.286 − 0.957i)13-s + (−0.835 − 0.549i)14-s + (−0.396 − 0.918i)15-s + (0.173 + 0.984i)16-s + (−0.939 − 0.342i)17-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)2-s + (0.396 − 0.918i)3-s + (0.766 + 0.642i)4-s + (0.686 − 0.727i)5-s + (−0.686 + 0.727i)6-s + (0.973 + 0.230i)7-s + (−0.5 − 0.866i)8-s + (−0.686 − 0.727i)9-s + (−0.893 + 0.448i)10-s + (−0.893 − 0.448i)11-s + (0.893 − 0.448i)12-s + (−0.286 − 0.957i)13-s + (−0.835 − 0.549i)14-s + (−0.396 − 0.918i)15-s + (0.173 + 0.984i)16-s + (−0.939 − 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.389 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.389 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.389 + 0.921i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (102, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.389 + 0.921i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3481002864 - 0.5250267430i\)
\(L(\frac12)\) \(\approx\) \(-0.3481002864 - 0.5250267430i\)
\(L(1)\) \(\approx\) \(0.5750657538 - 0.5152406681i\)
\(L(1)\) \(\approx\) \(0.5750657538 - 0.5152406681i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.939 - 0.342i)T \)
3 \( 1 + (0.396 - 0.918i)T \)
5 \( 1 + (0.686 - 0.727i)T \)
7 \( 1 + (0.973 + 0.230i)T \)
11 \( 1 + (-0.893 - 0.448i)T \)
13 \( 1 + (-0.286 - 0.957i)T \)
17 \( 1 + (-0.939 - 0.342i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.766 - 0.642i)T \)
29 \( 1 + (-0.893 + 0.448i)T \)
31 \( 1 + (-0.396 - 0.918i)T \)
41 \( 1 + (0.939 + 0.342i)T \)
43 \( 1 + (-0.766 - 0.642i)T \)
47 \( 1 + (-0.973 + 0.230i)T \)
53 \( 1 + (-0.893 + 0.448i)T \)
59 \( 1 + (0.396 + 0.918i)T \)
61 \( 1 + (0.835 + 0.549i)T \)
67 \( 1 + (0.835 - 0.549i)T \)
71 \( 1 + (-0.939 + 0.342i)T \)
73 \( 1 + (-0.0581 - 0.998i)T \)
79 \( 1 + (-0.0581 - 0.998i)T \)
83 \( 1 + (0.973 + 0.230i)T \)
89 \( 1 + (-0.597 - 0.802i)T \)
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

−18.96847486193613706685539821194, −18.02287935662766766855702568318, −17.517171522011900475368962138866, −17.13005071734643850260217296039, −16.1985935606608769792604162679, −15.52263929931827206965308466381, −14.83867485639958381340483029811, −14.58792838263591564344813187898, −13.73865092156324657080836330854, −13.00481209451071727654013471985, −11.47702572116745691105371275083, −11.051608196038673442963343650990, −10.69993026195963601980911081735, −9.75370411158138131795031940747, −9.38975603788469225771822678992, −8.58995919644265701083214442985, −7.91375874480422577622595401399, −7.08236960793633084053338919575, −6.55941702831127663703863660353, −5.32843846484131554511922293511, −5.0243504629564506538763294607, −4.010543558099553342390698703266, −2.820508781681223759998273835178, −2.17811633590417640579786678436, −1.64649519253893727998063237168, 0.21088605840162776480161589663, 1.11630573192987509710793233391, 1.94147185992791840732381077063, 2.41153275058935181257745948785, 3.207967735851457884454998263014, 4.4664646388835215898888833916, 5.491855091315754540430553845817, 6.02277077881852476892218700902, 7.03697040321369200830775355781, 7.827889326250558276635845061584, 8.26074283484550659176420702848, 8.82311917748905147081347392357, 9.482282649629572133829154749558, 10.4498195065676353280795207916, 11.0523781851140174154331521985, 11.78609171134065915546948871525, 12.74577349628414221295670320036, 12.89952690977500472642700860526, 13.67700936670037987925715996068, 14.727937944674753223087643529002, 15.18183903844300221966478652349, 16.2644272462045961905644800429, 16.885494278422648227893732481148, 17.6170292427139951603971462201, 18.003562776080848443545074573359

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