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Properties

Label 1-4033-4033.1310-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.636 + 0.771i$
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 + (0.5 − 0.866i)3-s + (0.766 − 0.642i)4-s + (−0.866 + 0.5i)5-s + (−0.173 + 0.984i)6-s + (−0.5 − 0.866i)7-s + (−0.5 + 0.866i)8-s + (−0.5 − 0.866i)9-s + (0.642 − 0.766i)10-s + (−0.642 − 0.766i)11-s + (−0.173 − 0.984i)12-s + (−0.5 + 0.866i)13-s + (0.766 + 0.642i)14-s + i·15-s + (0.173 − 0.984i)16-s + (−0.766 − 0.642i)17-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)2-s + (0.5 − 0.866i)3-s + (0.766 − 0.642i)4-s + (−0.866 + 0.5i)5-s + (−0.173 + 0.984i)6-s + (−0.5 − 0.866i)7-s + (−0.5 + 0.866i)8-s + (−0.5 − 0.866i)9-s + (0.642 − 0.766i)10-s + (−0.642 − 0.766i)11-s + (−0.173 − 0.984i)12-s + (−0.5 + 0.866i)13-s + (0.766 + 0.642i)14-s + i·15-s + (0.173 − 0.984i)16-s + (−0.766 − 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.636 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.636 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.636 + 0.771i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1310, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.636 + 0.771i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1099865609 + 0.05186107376i\)
\(L(\frac12)\) \(\approx\) \(0.1099865609 + 0.05186107376i\)
\(L(1)\) \(\approx\) \(0.4644931751 - 0.1483501813i\)
\(L(1)\) \(\approx\) \(0.4644931751 - 0.1483501813i\)

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 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.642 - 0.766i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (-0.766 - 0.642i)T \)
19 \( 1 + (0.173 + 0.984i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.984 - 0.173i)T \)
31 \( 1 + (-0.642 - 0.766i)T \)
41 \( 1 + (-0.984 + 0.173i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 + (-0.642 - 0.766i)T \)
53 \( 1 + (-0.342 + 0.939i)T \)
59 \( 1 + T \)
61 \( 1 - iT \)
67 \( 1 + (0.342 + 0.939i)T \)
71 \( 1 + (0.939 + 0.342i)T \)
73 \( 1 + (0.939 + 0.342i)T \)
79 \( 1 + (-0.939 + 0.342i)T \)
83 \( 1 + (-0.173 - 0.984i)T \)
89 \( 1 + (-0.984 + 0.173i)T \)
97 \( 1 + (-0.984 - 0.173i)T \)
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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

−18.41590180639203185493219187144, −17.704832898003042348993898581902, −17.00759696362773962968062898431, −16.22761207049160281864303754425, −15.63173099270780567158938489346, −15.23451760247630095006506686099, −14.86948262906441022647206773339, −13.087383300333629983277171111596, −12.98966421020028550714657528299, −12.07439872839327121265134952081, −11.27658335911582349166033178202, −10.77609139685399866285325770504, −9.84935414536901410224453761778, −9.39547970618921823441251793044, −8.71406243207043159794637816881, −8.16793353907860106262642472929, −7.47578073499186383652115368579, −6.71852639748416466334566927117, −5.34409841747215026618526496270, −4.94568836822495534180166035293, −3.78906468684013483654360444022, −3.19883219029207783493759450267, −2.50620791823933359528561851353, −1.66222787067171609662842330248, −0.07399794821956828307691747309, 0.54747686918357800513225302838, 1.72242098185217314409231981472, 2.556546052650916154136090189172, 3.28833163633276911400676498157, 4.09924367389092157678583586910, 5.35116878658240380489978219398, 6.37634755898442535053471753565, 6.94348664956826069867112340750, 7.30326138751760385293984004217, 8.1050346597650352936311480337, 8.56913761763434027630284060784, 9.49988945771665604594377286301, 10.141956985331964412793121777589, 11.13276294743929006613743980940, 11.41820958504793854191663746910, 12.32829572774965116116179311251, 13.179665144679080883222824070, 13.955341408737136251552271101485, 14.51180620212975056116979467220, 15.201704707401273708404051428594, 15.96733857022492044293520552947, 16.70400897064751688698358090136, 17.03561867252257107262242052949, 18.30822196801670602762390579220, 18.5709478452950721641976297799

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