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Properties

Label 1-4033-4033.135-r1-0-0
Degree $1$
Conductor $4033$
Sign $-0.488 - 0.872i$
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  + i·2-s + (−0.597 − 0.802i)3-s − 4-s + (0.230 − 0.973i)5-s + (0.802 − 0.597i)6-s + (−0.286 − 0.957i)7-s − i·8-s + (−0.286 + 0.957i)9-s + (0.973 + 0.230i)10-s + (−0.973 + 0.230i)11-s + (0.597 + 0.802i)12-s + (0.230 − 0.973i)13-s + (0.957 − 0.286i)14-s + (−0.918 + 0.396i)15-s + 16-s + (−0.866 + 0.5i)17-s + ⋯
L(s)  = 1  + i·2-s + (−0.597 − 0.802i)3-s − 4-s + (0.230 − 0.973i)5-s + (0.802 − 0.597i)6-s + (−0.286 − 0.957i)7-s − i·8-s + (−0.286 + 0.957i)9-s + (0.973 + 0.230i)10-s + (−0.973 + 0.230i)11-s + (0.597 + 0.802i)12-s + (0.230 − 0.973i)13-s + (0.957 − 0.286i)14-s + (−0.918 + 0.396i)15-s + 16-s + (−0.866 + 0.5i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.488 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.488 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.488 - 0.872i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (135, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ -0.488 - 0.872i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4205532719 - 0.7176730312i\)
\(L(\frac12)\) \(\approx\) \(0.4205532719 - 0.7176730312i\)
\(L(1)\) \(\approx\) \(0.6938979598 - 0.06118256767i\)
\(L(1)\) \(\approx\) \(0.6938979598 - 0.06118256767i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + iT \)
3 \( 1 + (-0.597 - 0.802i)T \)
5 \( 1 + (0.230 - 0.973i)T \)
7 \( 1 + (-0.286 - 0.957i)T \)
11 \( 1 + (-0.973 + 0.230i)T \)
13 \( 1 + (0.230 - 0.973i)T \)
17 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (-0.342 + 0.939i)T \)
23 \( 1 + (0.984 - 0.173i)T \)
29 \( 1 + (0.957 - 0.286i)T \)
31 \( 1 + (-0.448 + 0.893i)T \)
41 \( 1 + (-0.173 + 0.984i)T \)
43 \( 1 + (0.342 + 0.939i)T \)
47 \( 1 + (0.597 - 0.802i)T \)
53 \( 1 + (-0.993 + 0.116i)T \)
59 \( 1 + (0.918 - 0.396i)T \)
61 \( 1 + (0.549 - 0.835i)T \)
67 \( 1 + (-0.597 + 0.802i)T \)
71 \( 1 + T \)
73 \( 1 + (0.286 - 0.957i)T \)
79 \( 1 + (0.918 - 0.396i)T \)
83 \( 1 + (-0.0581 + 0.998i)T \)
89 \( 1 + (-0.727 - 0.686i)T \)
97 \( 1 + (-0.448 - 0.893i)T \)
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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

−18.71496767920242426419929098173, −17.8539477356291343246915110367, −17.51476562759585554138726062649, −16.51493080722304043398539517228, −15.56302390193343339489670062665, −15.3179727302638750645663102353, −14.38207226697130733875698325445, −13.67059177020763885656275908674, −12.96204898620450608911829626186, −12.137435568129667396639957320439, −11.373800234696738412806299650541, −11.02907557041844809432951018335, −10.46310684380481579823261086788, −9.58088486849182890800715953028, −9.11920962564871074014787481728, −8.51921319997504791544193293251, −7.16722484255898763758907265845, −6.41692461485236303425238566761, −5.59403341362745092867530165221, −4.99433650561057678574505297681, −4.199878614589337240346243302961, −3.31975671018523744905951499488, −2.60672447246504732040228257258, −2.15644352277350474181270975554, −0.6553773672929517565729461892, 0.243870964225696381583334731621, 0.8515953040971835907895657564, 1.67904783638917238850323172502, 2.98311177818481730476624534778, 4.10690172374876205423642484942, 4.88137965710350509214589732818, 5.34531845250987312140668108791, 6.21506072766489404960835879196, 6.71395896174520755793520290283, 7.63602282675139108118164761617, 8.12066655399182115738612493176, 8.651171310125419909809748588766, 9.79747642275453962542803424848, 10.40377598092628935855102748499, 11.0508993748595399257902203424, 12.448441031802595058138712513799, 12.73703084827384883172736535189, 13.26579082505258939772182686394, 13.756771251147913912121232770036, 14.66981336236817935734762360611, 15.62721754337429688727065323359, 16.17718363123818570723207100984, 16.75464699360982721393951047309, 17.41540536869945087288080317516, 17.762710890032800907902680585134

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