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Properties

Label 1-4033-4033.2076-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.961 + 0.276i$
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 + (0.973 + 0.230i)3-s + (−0.5 + 0.866i)4-s + (0.0581 − 0.998i)5-s + (0.286 + 0.957i)6-s + (0.893 − 0.448i)7-s − 8-s + (0.893 + 0.448i)9-s + (0.893 − 0.448i)10-s + (0.893 + 0.448i)11-s + (−0.686 + 0.727i)12-s + (0.0581 − 0.998i)13-s + (0.835 + 0.549i)14-s + (0.286 − 0.957i)15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (0.973 + 0.230i)3-s + (−0.5 + 0.866i)4-s + (0.0581 − 0.998i)5-s + (0.286 + 0.957i)6-s + (0.893 − 0.448i)7-s − 8-s + (0.893 + 0.448i)9-s + (0.893 − 0.448i)10-s + (0.893 + 0.448i)11-s + (−0.686 + 0.727i)12-s + (0.0581 − 0.998i)13-s + (0.835 + 0.549i)14-s + (0.286 − 0.957i)15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.961 + 0.276i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2076, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.961 + 0.276i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.961076891 + 0.5584351187i\)
\(L(\frac12)\) \(\approx\) \(3.961076891 + 0.5584351187i\)
\(L(1)\) \(\approx\) \(2.055076557 + 0.5615275254i\)
\(L(1)\) \(\approx\) \(2.055076557 + 0.5615275254i\)

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 \( 1 + (0.973 + 0.230i)T \)
5 \( 1 + (0.0581 - 0.998i)T \)
7 \( 1 + (0.893 - 0.448i)T \)
11 \( 1 + (0.893 + 0.448i)T \)
13 \( 1 + (0.0581 - 0.998i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.173 - 0.984i)T \)
23 \( 1 + (0.939 + 0.342i)T \)
29 \( 1 + (0.835 + 0.549i)T \)
31 \( 1 + (-0.597 + 0.802i)T \)
41 \( 1 + (0.173 - 0.984i)T \)
43 \( 1 + (-0.766 - 0.642i)T \)
47 \( 1 + (-0.286 + 0.957i)T \)
53 \( 1 + (-0.286 - 0.957i)T \)
59 \( 1 + (0.286 - 0.957i)T \)
61 \( 1 + (-0.597 - 0.802i)T \)
67 \( 1 + (-0.686 - 0.727i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (-0.835 + 0.549i)T \)
79 \( 1 + (-0.973 + 0.230i)T \)
83 \( 1 + (0.597 + 0.802i)T \)
89 \( 1 + (-0.893 + 0.448i)T \)
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

−18.639031794645591949652180035667, −18.22307027884226871739161858509, −17.29125919026197150687414902647, −16.33691265481319393367018474506, −15.02980671531065259848706057545, −14.830756914742764201588263469156, −14.42803214635898552595714228381, −13.7333352534567141226901914821, −13.13641640262139044103776493125, −12.00979497644900052673227279160, −11.79650876321006834093282997251, −10.90506585796280685806793808358, −10.24590854796611419632561211621, −9.48420639768174624579951477022, −8.78767288197631877992672083022, −8.1758134344624733120895201836, −7.23457717349299762185818592425, −6.30117068543938308701505327912, −5.87597189969818714090621816051, −4.504711146059951884481290266437, −4.047820149251087900924075148485, −3.23024952368897551632644796977, −2.606348586318682943157910666269, −1.67585961649249067683955575094, −1.38075300857458108243878686921, 0.839876776126782526888188799503, 1.71057380505339162575276070570, 2.9040761250669837366288525892, 3.590037672536356506955899107661, 4.477218108704576846132465850, 4.9277760419311611406175492704, 5.46134260311086593711088361664, 6.87086081506974675859468609174, 7.27552144975397221626614703546, 8.093785328820334796595171933178, 8.636898743798637195241293429702, 9.193216169171554584127930116381, 9.886919389451259897537775957974, 10.97137713735501677941542823415, 11.89352738791134866971113604961, 12.6699818127555897098621709234, 13.14054859989231689571384134592, 14.01831749472760160034527540186, 14.27610410400775522086680148790, 15.13292659369777178873186020152, 15.641512148107777509247596154364, 16.31775845915331096184927866814, 17.11578806740396458962386734489, 17.58572437302895179694818901174, 18.2242024467198817047399904732

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