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Properties

Label 1-4033-4033.951-r1-0-0
Degree $1$
Conductor $4033$
Sign $-0.642 - 0.766i$
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  + (−0.984 + 0.173i)2-s + (−0.0581 − 0.998i)3-s + (0.939 − 0.342i)4-s + (0.597 − 0.802i)5-s + (0.230 + 0.973i)6-s + (0.396 + 0.918i)7-s + (−0.866 + 0.5i)8-s + (−0.993 + 0.116i)9-s + (−0.448 + 0.893i)10-s + (−0.448 − 0.893i)11-s + (−0.396 − 0.918i)12-s + (0.918 − 0.396i)13-s + (−0.549 − 0.835i)14-s + (−0.835 − 0.549i)15-s + (0.766 − 0.642i)16-s + (0.642 + 0.766i)17-s + ⋯
L(s)  = 1  + (−0.984 + 0.173i)2-s + (−0.0581 − 0.998i)3-s + (0.939 − 0.342i)4-s + (0.597 − 0.802i)5-s + (0.230 + 0.973i)6-s + (0.396 + 0.918i)7-s + (−0.866 + 0.5i)8-s + (−0.993 + 0.116i)9-s + (−0.448 + 0.893i)10-s + (−0.448 − 0.893i)11-s + (−0.396 − 0.918i)12-s + (0.918 − 0.396i)13-s + (−0.549 − 0.835i)14-s + (−0.835 − 0.549i)15-s + (0.766 − 0.642i)16-s + (0.642 + 0.766i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.642 - 0.766i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (951, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ -0.642 - 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6357289980 - 1.361989110i\)
\(L(\frac12)\) \(\approx\) \(0.6357289980 - 1.361989110i\)
\(L(1)\) \(\approx\) \(0.7474162436 - 0.3260052082i\)
\(L(1)\) \(\approx\) \(0.7474162436 - 0.3260052082i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.984 + 0.173i)T \)
3 \( 1 + (-0.0581 - 0.998i)T \)
5 \( 1 + (0.597 - 0.802i)T \)
7 \( 1 + (0.396 + 0.918i)T \)
11 \( 1 + (-0.448 - 0.893i)T \)
13 \( 1 + (0.918 - 0.396i)T \)
17 \( 1 + (0.642 + 0.766i)T \)
19 \( 1 + (-0.642 - 0.766i)T \)
23 \( 1 + (0.984 + 0.173i)T \)
29 \( 1 + (0.0581 + 0.998i)T \)
31 \( 1 + (0.993 - 0.116i)T \)
41 \( 1 + iT \)
43 \( 1 + (-0.766 + 0.642i)T \)
47 \( 1 + (-0.727 - 0.686i)T \)
53 \( 1 + (0.116 - 0.993i)T \)
59 \( 1 + (-0.549 + 0.835i)T \)
61 \( 1 + (0.286 - 0.957i)T \)
67 \( 1 + (0.802 - 0.597i)T \)
71 \( 1 + (-0.173 + 0.984i)T \)
73 \( 1 + (0.893 - 0.448i)T \)
79 \( 1 + (-0.918 - 0.396i)T \)
83 \( 1 + (0.0581 + 0.998i)T \)
89 \( 1 + (-0.286 + 0.957i)T \)
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

−18.61090633818337658928060776725, −17.668708254648890551815587094165, −17.24254624613892441761895434672, −16.73510149234390253908483909540, −15.88204354258171761707003420474, −15.310351985014952609236949276126, −14.58741714325806256790609031880, −13.979779389806118435534716356159, −13.143583735004803829672713273411, −12.005612564794881249650593322419, −11.33455959927926292035940741478, −10.69778792658003609039584920040, −10.22244662754412175795287800771, −9.79528746125940359261381483028, −9.00834047657213366564555769873, −8.16937065060520816924613877161, −7.45768002300419053167370176043, −6.697478850248123489520941815962, −6.02564601588430013069425538197, −5.05947068638223354034379623104, −4.12975520129800712451893029515, −3.38756141200027390709881490937, −2.607206638915355625636968839096, −1.7713367754872422113397388230, −0.78848959250254522743910833085, 0.38276179316417554397113700679, 1.186813579907283989767019321183, 1.66414606484661091135162207172, 2.621889514177265002270300679901, 3.23500046035060486303167958359, 5.041001106029515262003933181799, 5.49988307912097062963958879396, 6.23993365993927577601995468817, 6.676330202124118728024337939733, 7.98031715257074282250059359116, 8.34916227622616243991498396415, 8.69762143757458919263650245223, 9.49399314151423345160241760166, 10.50939306115616384735710575118, 11.203670553725370319569808370357, 11.72350645714895777069220693878, 12.72848591830908053064232788148, 13.01027864944453124980859414368, 13.9120767238184649234129898679, 14.76986939213393651805113086554, 15.43649912584105671604728957405, 16.2868832753737490286973445899, 16.8722040799084195521008383086, 17.47229516667260337589272470044, 18.17918985948752317767673468403

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