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Properties

Label 1-4033-4033.55-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.219 - 0.975i$
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  + (−0.766 + 0.642i)2-s + (0.5 − 0.866i)3-s + (0.173 − 0.984i)4-s − i·5-s + (0.173 + 0.984i)6-s + 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 + (−0.766 − 0.642i)12-s + (0.5 − 0.866i)13-s + (−0.766 + 0.642i)14-s + (−0.866 − 0.5i)15-s + (−0.939 − 0.342i)16-s + (0.173 + 0.984i)17-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)2-s + (0.5 − 0.866i)3-s + (0.173 − 0.984i)4-s − i·5-s + (0.173 + 0.984i)6-s + 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 + (−0.766 − 0.642i)12-s + (0.5 − 0.866i)13-s + (−0.766 + 0.642i)14-s + (−0.866 − 0.5i)15-s + (−0.939 − 0.342i)16-s + (0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.219 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.219 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.219 - 0.975i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.219 - 0.975i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9883708566 - 1.236068206i\)
\(L(\frac12)\) \(\approx\) \(0.9883708566 - 1.236068206i\)
\(L(1)\) \(\approx\) \(0.9426263028 - 0.3306883468i\)
\(L(1)\) \(\approx\) \(0.9426263028 - 0.3306883468i\)

Euler product

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

−18.50861369041528892810924115671, −18.23003630559317247505194691230, −17.50396105967313694428254179820, −16.568788106030670182811751744031, −15.88517463054484376697113049244, −15.58392270923191974230288614933, −14.40166240999837406079891790428, −13.87806239807424889791266488693, −13.585809674085192454120367483437, −12.03743355090854386774287908537, −11.47616789411891224403074407148, −11.07421732442434541188418179112, −10.30771721759496758274993357678, −9.864230876064974738164608307928, −8.94651263082946815820881189488, −8.41396687248772515907594368510, −7.64440323967174011586702020935, −7.15559233491803003542231523464, −5.92595331596829551023635843173, −5.04073755229654160044617845395, −4.138747056993938918854924134818, −3.413221957912888836493750992102, −2.80111500040794428617065269691, −2.11205704164629239420230337101, −1.102731518660613784301812197303, 0.61024836345613934969590979611, 1.269393611861195255308164661099, 1.97523328031734798043720168679, 2.804002996694916189661783596011, 4.2542736417454743239325433624, 4.921276854473647545794616283420, 5.77833587409862623726692052016, 6.29453564048108664344875370416, 7.58787722841891352593714443846, 7.736146666790259354796829822535, 8.4223462844812315906670822163, 8.87700623907322072450036165745, 9.848961948857177809122304217135, 10.467802227124117559289505364423, 11.456198380945609133678668032182, 12.157587262968931811294038240046, 12.89066732248420458640951005310, 13.57146224026118852575512252788, 14.23941651247067646810601538579, 15.04130392627094647632927507585, 15.48385436197278951929153986069, 16.2898496668061649519061350110, 17.17180170669018172987658223383, 17.81586054653666529125781025378, 17.94254924288099568638813585503

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