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Properties

Label 1-4033-4033.1699-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.498 - 0.867i$
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  + (−0.173 − 0.984i)2-s + (−0.939 + 0.342i)3-s + (−0.939 + 0.342i)4-s + (0.173 − 0.984i)5-s + (0.5 + 0.866i)6-s + (0.173 − 0.984i)7-s + (0.5 + 0.866i)8-s + (0.766 − 0.642i)9-s − 10-s − 11-s + (0.766 − 0.642i)12-s + (−0.766 − 0.642i)13-s − 14-s + (0.173 + 0.984i)15-s + (0.766 − 0.642i)16-s + (0.939 + 0.342i)17-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)2-s + (−0.939 + 0.342i)3-s + (−0.939 + 0.342i)4-s + (0.173 − 0.984i)5-s + (0.5 + 0.866i)6-s + (0.173 − 0.984i)7-s + (0.5 + 0.866i)8-s + (0.766 − 0.642i)9-s − 10-s − 11-s + (0.766 − 0.642i)12-s + (−0.766 − 0.642i)13-s − 14-s + (0.173 + 0.984i)15-s + (0.766 − 0.642i)16-s + (0.939 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.498 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.498 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.498 - 0.867i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1699, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.498 - 0.867i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4378370501 - 0.7565433829i\)
\(L(\frac12)\) \(\approx\) \(0.4378370501 - 0.7565433829i\)
\(L(1)\) \(\approx\) \(0.5387502761 - 0.4027950026i\)
\(L(1)\) \(\approx\) \(0.5387502761 - 0.4027950026i\)

Euler product

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

−18.6519864009712756733624281584, −17.89325239852133656054236322687, −17.4463126432434418230655792248, −16.728404177777879570129950792826, −15.94193397658705934078995100280, −15.33615718697744193256353420481, −14.859025182788270022974161724627, −14.01793942829231048125317377737, −13.27205008468013944269499332383, −12.68859565862326595660360332065, −11.69183699053004226054342868103, −11.2351594794622216669208288012, −10.2986273267757529023862719781, −9.68975801628677196833862663315, −9.033483216608895363364868404709, −7.76455431692668442683040069148, −7.49754067654543791515659380434, −6.79930524292923889866626064767, −5.94308840557788935450039286523, −5.49420561718395655430831303033, −4.92795897220775590937524274295, −3.91829857473118121098907301563, −2.63287285651494064095907933485, −2.024849514062393245514794051301, −0.627290772979074330877647007951, 0.56673355921436773694240356976, 1.1144863496239900374990623180, 2.070327016455950635428873332048, 3.252005861543506651380693553351, 4.01136007264176852279785693495, 4.72004090426920665599384670943, 5.28580733240812960441015284336, 5.8404418464025539461913112321, 7.28102471426699324406144032255, 7.80915126777037512452766974799, 8.63949008775586469701611543758, 9.5627724189683015791049428442, 10.17666132097660822128293170535, 10.59623879864881775851095422761, 11.16725025626594547219343943387, 12.348795367632348771228936764051, 12.55014075370225778132017705142, 12.981258650769938859023657994585, 14.066518783647010312350101026247, 14.65535791795775624860575610775, 15.808732876709057810358590394724, 16.638406429551560842074492522797, 16.83048649471727579405488337194, 17.59177997358457038570439587230, 18.117017235141785950208267951645

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