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Properties

Label 1-4033-4033.930-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.958 - 0.283i$
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 + (−0.893 + 0.448i)3-s + (−0.5 + 0.866i)4-s + (−0.116 − 0.993i)5-s + (0.835 + 0.549i)6-s + (0.597 + 0.802i)7-s + 8-s + (0.597 − 0.802i)9-s + (−0.802 + 0.597i)10-s + (0.802 + 0.597i)11-s + (0.0581 − 0.998i)12-s + (−0.993 + 0.116i)13-s + (0.396 − 0.918i)14-s + (0.549 + 0.835i)15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.893 + 0.448i)3-s + (−0.5 + 0.866i)4-s + (−0.116 − 0.993i)5-s + (0.835 + 0.549i)6-s + (0.597 + 0.802i)7-s + 8-s + (0.597 − 0.802i)9-s + (−0.802 + 0.597i)10-s + (0.802 + 0.597i)11-s + (0.0581 − 0.998i)12-s + (−0.993 + 0.116i)13-s + (0.396 − 0.918i)14-s + (0.549 + 0.835i)15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.958 - 0.283i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (930, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.958 - 0.283i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7231506875 - 0.1047879708i\)
\(L(\frac12)\) \(\approx\) \(0.7231506875 - 0.1047879708i\)
\(L(1)\) \(\approx\) \(0.5857980440 - 0.1547320550i\)
\(L(1)\) \(\approx\) \(0.5857980440 - 0.1547320550i\)

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 \( 1 + (-0.893 + 0.448i)T \)
5 \( 1 + (-0.116 - 0.993i)T \)
7 \( 1 + (0.597 + 0.802i)T \)
11 \( 1 + (0.802 + 0.597i)T \)
13 \( 1 + (-0.993 + 0.116i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.939 - 0.342i)T \)
23 \( 1 + (-0.766 + 0.642i)T \)
29 \( 1 + (0.918 + 0.396i)T \)
31 \( 1 + (-0.957 - 0.286i)T \)
41 \( 1 + (0.342 + 0.939i)T \)
43 \( 1 + (-0.984 - 0.173i)T \)
47 \( 1 + (-0.549 - 0.835i)T \)
53 \( 1 + (0.549 - 0.835i)T \)
59 \( 1 + (-0.835 + 0.549i)T \)
61 \( 1 + (0.957 - 0.286i)T \)
67 \( 1 + (0.998 - 0.0581i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (-0.396 - 0.918i)T \)
79 \( 1 + (0.893 + 0.448i)T \)
83 \( 1 + (0.286 + 0.957i)T \)
89 \( 1 + (0.802 - 0.597i)T \)
97 \( 1 + (0.957 - 0.286i)T \)
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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

−18.36817123486009465897913075744, −17.56470731562683550834440779668, −17.25762360142076052261820101565, −16.6796820521455129765313702031, −16.0007140402764967480419766926, −15.02913832437658896056558071798, −14.34644059330510675842747645536, −14.15723835919537568644623043327, −13.13505845794578799692732031404, −12.25004368572294535603851018988, −11.48343823644257442886793833645, −10.61309901257604706044009162099, −10.46727843170398785498522317960, −9.661479158590504249010497268857, −8.37890906611977734102707532074, −7.91119119461602213428863960784, −7.19789648529251884648291617599, −6.57208749842684129577114526404, −6.12162583691522530190409650012, −5.28553901649615676448789688838, −4.361345581912901072329734677336, −3.78324318262742845759360093580, −2.24041616634876292510430093971, −1.486109887025945883146260450194, −0.47433999861435060973276844286, 0.61424582830361134932365543305, 1.61706314287630881698675065089, 2.18795893722402951118887984733, 3.45775849604450486225563589575, 4.29145451465861300076499241963, 4.91663561600242452907478431942, 5.28189056527967058137200377293, 6.4842441209006872861819975035, 7.382471427515075646463565448909, 8.20509182928752812869036782273, 8.98229977901700244962790522683, 9.57526998955119500628150331239, 9.96288014746997590157730742986, 11.0365304143059762730553493960, 11.72482629594518321204381396430, 12.0696683515814137907657771352, 12.50342571983549061021262181146, 13.34055549132684975317315547795, 14.44066639211378869000896858816, 15.105422018794574187163787618742, 16.00110665484952672878653997848, 16.685786726074503111531314019042, 17.07631994456378116576880487436, 17.88707719729447779925212620315, 18.13631968418509878311118645813

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