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Properties

Label 1-4033-4033.1698-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.106 + 0.994i$
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  + (−0.939 − 0.342i)2-s + (0.173 − 0.984i)3-s + (0.766 + 0.642i)4-s + (0.766 + 0.642i)5-s + (−0.5 + 0.866i)6-s + (0.766 + 0.642i)7-s + (−0.5 − 0.866i)8-s + (−0.939 − 0.342i)9-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (0.766 − 0.642i)12-s + (−0.939 + 0.342i)13-s + (−0.5 − 0.866i)14-s + (0.766 − 0.642i)15-s + (0.173 + 0.984i)16-s + (0.766 − 0.642i)17-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)2-s + (0.173 − 0.984i)3-s + (0.766 + 0.642i)4-s + (0.766 + 0.642i)5-s + (−0.5 + 0.866i)6-s + (0.766 + 0.642i)7-s + (−0.5 − 0.866i)8-s + (−0.939 − 0.342i)9-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (0.766 − 0.642i)12-s + (−0.939 + 0.342i)13-s + (−0.5 − 0.866i)14-s + (0.766 − 0.642i)15-s + (0.173 + 0.984i)16-s + (0.766 − 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.106 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.106 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.106 + 0.994i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1698, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.106 + 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5144223924 + 0.5725463866i\)
\(L(\frac12)\) \(\approx\) \(0.5144223924 + 0.5725463866i\)
\(L(1)\) \(\approx\) \(0.7616811183 - 0.04671433373i\)
\(L(1)\) \(\approx\) \(0.7616811183 - 0.04671433373i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.939 - 0.342i)T \)
3 \( 1 + (0.173 - 0.984i)T \)
5 \( 1 + (0.766 + 0.642i)T \)
7 \( 1 + (0.766 + 0.642i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.939 + 0.342i)T \)
17 \( 1 + (0.766 - 0.642i)T \)
19 \( 1 + (-0.939 + 0.342i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + T \)
31 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.766 + 0.642i)T \)
43 \( 1 + T \)
47 \( 1 + T \)
53 \( 1 + (-0.939 - 0.342i)T \)
59 \( 1 + (0.766 - 0.642i)T \)
61 \( 1 + (-0.939 + 0.342i)T \)
67 \( 1 + (-0.939 + 0.342i)T \)
71 \( 1 + (-0.939 + 0.342i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (-0.939 + 0.342i)T \)
83 \( 1 + (0.173 + 0.984i)T \)
89 \( 1 + (-0.939 - 0.342i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−17.94043043678673789786771321053, −17.467014342463106804254998942056, −16.922588749265666627474595135556, −16.42869116318057200458663021946, −15.80239383033375525144965032511, −14.94076324694143937612255169257, −14.24852372436225286491455085574, −13.989191583695159955005964331105, −12.74595408718464097009652085343, −11.98445560694418287217184424575, −10.90119681097558389699085040503, −10.43522811479989035820586812691, −10.17898736885328692846082476477, −9.14397616418842416321883543368, −8.61555709874725767039678597188, −8.09290632891254104070576901410, −7.33369189712304827546033241879, −6.12099167943605738124092453341, −5.643679002736785347835575399994, −4.86415158344631654657777862357, −4.24704230303682472596947159395, −2.9235707337905889406845248603, −2.28880123333265260745941155449, −1.26469596566377027698250108329, −0.27639413133490645367239268319, 1.2562131010330238377871794120, 1.97241792388308575969585238033, 2.44084032780701120013156215941, 2.99730362513494213097844330360, 4.373323537963532321107498908719, 5.54331652412804230321869779670, 6.09792362642699028485849403504, 7.05082702768644775789655553871, 7.54644005214450112755385943925, 8.04521219659376361031889025708, 9.03977946442798885014108027132, 9.58850062407614744521040171701, 10.27267134302651261435559540149, 11.11729395805469112121906349128, 11.797881338035902042741618463202, 12.36150820197061668161352249895, 12.94710635752818598107675954370, 13.93968502756078315073448060467, 14.611146022981893041066010316033, 15.0763787026142261350398015058, 16.06397617703172733498336530687, 17.223144001450495410815696684804, 17.36983179855941425020830953644, 18.14189007379208773932326090131, 18.46609867537620608202793944533

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