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Properties

Label 1-4033-4033.2666-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.931 - 0.363i$
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 + (−0.597 + 0.802i)3-s + (−0.5 − 0.866i)4-s + (−0.230 − 0.973i)5-s + (−0.396 − 0.918i)6-s + (−0.286 + 0.957i)7-s + 8-s + (−0.286 − 0.957i)9-s + (0.957 + 0.286i)10-s + (−0.957 + 0.286i)11-s + (0.993 + 0.116i)12-s + (0.973 − 0.230i)13-s + (−0.686 − 0.727i)14-s + (0.918 + 0.396i)15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.597 + 0.802i)3-s + (−0.5 − 0.866i)4-s + (−0.230 − 0.973i)5-s + (−0.396 − 0.918i)6-s + (−0.286 + 0.957i)7-s + 8-s + (−0.286 − 0.957i)9-s + (0.957 + 0.286i)10-s + (−0.957 + 0.286i)11-s + (0.993 + 0.116i)12-s + (0.973 − 0.230i)13-s + (−0.686 − 0.727i)14-s + (0.918 + 0.396i)15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.931 - 0.363i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.931 - 0.363i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.931 - 0.363i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2666, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.931 - 0.363i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1117797580 + 0.5934103289i\)
\(L(\frac12)\) \(\approx\) \(-0.1117797580 + 0.5934103289i\)
\(L(1)\) \(\approx\) \(0.4463663781 + 0.3788234768i\)
\(L(1)\) \(\approx\) \(0.4463663781 + 0.3788234768i\)

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 \( 1 + (-0.597 + 0.802i)T \)
5 \( 1 + (-0.230 - 0.973i)T \)
7 \( 1 + (-0.286 + 0.957i)T \)
11 \( 1 + (-0.957 + 0.286i)T \)
13 \( 1 + (0.973 - 0.230i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (0.766 + 0.642i)T \)
23 \( 1 + (-0.173 + 0.984i)T \)
29 \( 1 + (-0.727 + 0.686i)T \)
31 \( 1 + (-0.549 + 0.835i)T \)
41 \( 1 + (0.642 + 0.766i)T \)
43 \( 1 + (0.342 - 0.939i)T \)
47 \( 1 + (-0.918 - 0.396i)T \)
53 \( 1 + (0.918 - 0.396i)T \)
59 \( 1 + (0.396 - 0.918i)T \)
61 \( 1 + (0.549 + 0.835i)T \)
67 \( 1 + (0.116 + 0.993i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (0.686 - 0.727i)T \)
79 \( 1 + (0.597 + 0.802i)T \)
83 \( 1 + (0.835 - 0.549i)T \)
89 \( 1 + (-0.957 - 0.286i)T \)
97 \( 1 + (0.549 + 0.835i)T \)
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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

−18.103355656762868608413451043788, −17.8951957637427699781520395611, −16.762102316769594566708212287887, −16.367330847793101987128283800097, −15.640033650510390187137731205410, −14.28736811522056868084001154449, −13.71717401180088803589151736329, −13.28286483109118161016859161834, −12.57075040861619501791329409991, −11.629064943756252910461989334912, −11.13355381821771437009783438701, −10.75588593745212648585057107915, −10.03209573877436567567521790022, −9.247565137782553996944226542, −8.01264012295544638133518713082, −7.70342313988821832108120579997, −7.04260205307202576574909098755, −6.3010003160595196809864985656, −5.35188671411076963091527728654, −4.34801852794288343539972267633, −3.51716769158477544812612884718, −2.772621542770837244201205680983, −2.11765992640172433714842460888, −0.925739547303421687133618372415, −0.32431347129651926637915228879, 0.98175053349621457331297288209, 1.82856154744233921578604333350, 3.39862093254308550812578998597, 3.97702092072288618981370443841, 5.154788362202265327496652915316, 5.458272899420396157511953616049, 5.837961166372239531392593654234, 6.88228668899163036454626974625, 7.93447782814391577745792122551, 8.41876693897045137528801333824, 9.13745450597460540235498042707, 9.70821076212601684845492538265, 10.3660537613475892979126905491, 11.16178883343618395847909010167, 11.9673267761406400809026508601, 12.76935398055360666339951269698, 13.27462635369484883723657019323, 14.4359141172628651452930530636, 15.11031918341577863611966458126, 15.72732176298524343199381417960, 16.12660812080672944534722713266, 16.53429465336192683257210684141, 17.43358551279065407523553992598, 18.059858385131905716333322087929, 18.50693978839381964378831510605

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