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Properties

Label 1-4033-4033.2075-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.600 - 0.799i$
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 + (0.5 − 0.866i)5-s + (−0.939 − 0.342i)6-s + (−0.5 + 0.866i)7-s + (−0.5 − 0.866i)8-s + (−0.5 + 0.866i)9-s + (−0.173 − 0.984i)10-s + (−0.173 + 0.984i)11-s + (−0.939 + 0.342i)12-s + (−0.5 − 0.866i)13-s + (0.173 + 0.984i)14-s − 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 + (0.5 − 0.866i)5-s + (−0.939 − 0.342i)6-s + (−0.5 + 0.866i)7-s + (−0.5 − 0.866i)8-s + (−0.5 + 0.866i)9-s + (−0.173 − 0.984i)10-s + (−0.173 + 0.984i)11-s + (−0.939 + 0.342i)12-s + (−0.5 − 0.866i)13-s + (0.173 + 0.984i)14-s − 15-s + (−0.939 − 0.342i)16-s + (0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.600 - 0.799i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2075, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.600 - 0.799i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.384067630 - 0.6910146956i\)
\(L(\frac12)\) \(\approx\) \(1.384067630 - 0.6910146956i\)
\(L(1)\) \(\approx\) \(0.9666715754 - 0.6989210285i\)
\(L(1)\) \(\approx\) \(0.9666715754 - 0.6989210285i\)

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

−18.358055274302885446069871502224, −17.558526092713789289529677303577, −16.98457078270996559510514436051, −16.50736543902052683997661207216, −15.850448635702031711244803099767, −15.21861807206149570082642669516, −14.44061210065656584893815072344, −13.80965507699343201828925563667, −13.54570720202439878799621122185, −12.42088778028433170707094344198, −11.57485524597558776608028631882, −11.102481802887752291407637933227, −10.3923732342342697098199872865, −9.57175142727485285986288783023, −9.01883278072320578909154608226, −7.83352602240140262438279674273, −7.12330247346161053447926155273, −6.36789768388664658705589680809, −6.02727186217037175843893644904, −5.112723941676714997388948710797, −4.34561511004632866802435661559, −3.67867283420644263119107346207, −3.02820899447591711238682402015, −2.22576395999443051878548814345, −0.41813425194418274753256373474, 0.82528223587568103627859461771, 1.78048270777096792131753107053, 2.27254023244069701924081145036, 3.01756924964407513823246882987, 4.340036835958914505394330196130, 4.91450728192658561627825256292, 5.66729756217112252272832992166, 6.15370049466671303936386810536, 6.79076617719744159319708886558, 7.96662601709079054793852675055, 8.61534960213907960347248704251, 9.55794883165902909118259175692, 10.279183725755056077245207907897, 10.77451325006253004763892914298, 11.99519889309268746026332551306, 12.40506578058231972835892679350, 12.791858106085373011978873712347, 13.05548220484104268604390529252, 14.29317691655551386841958383573, 14.64989498503097127391701036659, 15.68961856598201760122737541503, 16.210550295865125674444015890834, 17.21291362646542254315342709474, 17.80050099451819514844398955110, 18.32360119039001772173553714362

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