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Properties

Label 1-4033-4033.1437-r1-0-0
Degree $1$
Conductor $4033$
Sign $-0.978 + 0.206i$
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 + (0.993 − 0.116i)3-s + (0.939 + 0.342i)4-s + (−0.957 + 0.286i)5-s + (−0.998 − 0.0581i)6-s + (−0.686 + 0.727i)7-s + (−0.866 − 0.5i)8-s + (0.973 − 0.230i)9-s + (0.993 − 0.116i)10-s + (−0.993 − 0.116i)11-s + (0.973 + 0.230i)12-s + (−0.727 − 0.686i)13-s + (0.802 − 0.597i)14-s + (−0.918 + 0.396i)15-s + (0.766 + 0.642i)16-s + (−0.642 + 0.766i)17-s + ⋯
L(s)  = 1  + (−0.984 − 0.173i)2-s + (0.993 − 0.116i)3-s + (0.939 + 0.342i)4-s + (−0.957 + 0.286i)5-s + (−0.998 − 0.0581i)6-s + (−0.686 + 0.727i)7-s + (−0.866 − 0.5i)8-s + (0.973 − 0.230i)9-s + (0.993 − 0.116i)10-s + (−0.993 − 0.116i)11-s + (0.973 + 0.230i)12-s + (−0.727 − 0.686i)13-s + (0.802 − 0.597i)14-s + (−0.918 + 0.396i)15-s + (0.766 + 0.642i)16-s + (−0.642 + 0.766i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.978 + 0.206i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.978 + 0.206i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.978 + 0.206i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1437, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ -0.978 + 0.206i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01981906501 - 0.1902709391i\)
\(L(\frac12)\) \(\approx\) \(0.01981906501 - 0.1902709391i\)
\(L(1)\) \(\approx\) \(0.6436050509 - 0.06895537491i\)
\(L(1)\) \(\approx\) \(0.6436050509 - 0.06895537491i\)

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 \( 1 + (0.993 - 0.116i)T \)
5 \( 1 + (-0.957 + 0.286i)T \)
7 \( 1 + (-0.686 + 0.727i)T \)
11 \( 1 + (-0.993 - 0.116i)T \)
13 \( 1 + (-0.727 - 0.686i)T \)
17 \( 1 + (-0.642 + 0.766i)T \)
19 \( 1 + (-0.342 - 0.939i)T \)
23 \( 1 + (0.342 - 0.939i)T \)
29 \( 1 + (0.918 + 0.396i)T \)
31 \( 1 + (0.727 - 0.686i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.984 - 0.173i)T \)
47 \( 1 + (0.835 - 0.549i)T \)
53 \( 1 + (0.286 + 0.957i)T \)
59 \( 1 + (-0.918 + 0.396i)T \)
61 \( 1 + (0.549 - 0.835i)T \)
67 \( 1 + (-0.686 - 0.727i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
73 \( 1 + (0.993 + 0.116i)T \)
79 \( 1 + (0.230 + 0.973i)T \)
83 \( 1 + (0.597 + 0.802i)T \)
89 \( 1 + (-0.998 - 0.0581i)T \)
97 \( 1 + (0.230 - 0.973i)T \)
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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

−18.846050408675747111715026828160, −18.15215120818796211536511236236, −17.23974870539758299337792109716, −16.38673279016245033197338050700, −16.06987416624026508555893059605, −15.41188881602113715103652535615, −14.86069154526825823791532056957, −13.942697203121403326683743747238, −13.2855597525060554824673011067, −12.40724895099269148120028337971, −11.769523092805650558679585979399, −10.840763928504765340721765579106, −10.11928334204824902312606949043, −9.667779557117559138493007073295, −8.872328648969655595422521633328, −8.19543816517823384920198302906, −7.59578128915789540743903003356, −7.104908485685557996729891270708, −6.42966980613701205294104139979, −5.01171490564768992148172235868, −4.39687257609474708436031421665, −3.34662531989353755292771855695, −2.83932284499118243335154843490, −1.884396701557028461916987383057, −0.90594047590296732364336516729, 0.05183327176973022275695883362, 0.71876199247945538959702825183, 2.209131835747041211028512640154, 2.65114819723856751324480861857, 3.143996882912833174987522493940, 4.08967945502237908214778259257, 5.09744169867721057676618174271, 6.394439910400733531734235146623, 6.908827268407999653676236081885, 7.64348653321478823569922306265, 8.44177013292747201110331814498, 8.59277461462758302274601476726, 9.52345623100429715179784073423, 10.391810327097245433878631902443, 10.67955777852914713295560733449, 11.78015372044211008839164453892, 12.58756056434268083581229612473, 12.77890444282802536040888828080, 13.83045590961293439739894058194, 15.12359398227260225142936873024, 15.276430709823261116248237689911, 15.62465774159600261471112291640, 16.52638839770072486815095625238, 17.37600437064703567684843106588, 18.34760668857065421706368434160

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