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Properties

Label 1-4033-4033.225-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.989 - 0.141i$
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  + (0.5 − 0.866i)2-s + (0.973 + 0.230i)3-s + (−0.5 − 0.866i)4-s + (0.0581 − 0.998i)5-s + (0.686 − 0.727i)6-s + (0.893 − 0.448i)7-s − 8-s + (0.893 + 0.448i)9-s + (−0.835 − 0.549i)10-s + (−0.835 + 0.549i)11-s + (−0.286 − 0.957i)12-s + (0.0581 − 0.998i)13-s + (0.0581 − 0.998i)14-s + (0.286 − 0.957i)15-s + (−0.5 + 0.866i)16-s − 17-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (0.973 + 0.230i)3-s + (−0.5 − 0.866i)4-s + (0.0581 − 0.998i)5-s + (0.686 − 0.727i)6-s + (0.893 − 0.448i)7-s − 8-s + (0.893 + 0.448i)9-s + (−0.835 − 0.549i)10-s + (−0.835 + 0.549i)11-s + (−0.286 − 0.957i)12-s + (0.0581 − 0.998i)13-s + (0.0581 − 0.998i)14-s + (0.286 − 0.957i)15-s + (−0.5 + 0.866i)16-s − 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.989 - 0.141i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (225, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.989 - 0.141i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2144820649 - 3.020877902i\)
\(L(\frac12)\) \(\approx\) \(0.2144820649 - 3.020877902i\)
\(L(1)\) \(\approx\) \(1.293687261 - 1.276722971i\)
\(L(1)\) \(\approx\) \(1.293687261 - 1.276722971i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (0.973 + 0.230i)T \)
5 \( 1 + (0.0581 - 0.998i)T \)
7 \( 1 + (0.893 - 0.448i)T \)
11 \( 1 + (-0.835 + 0.549i)T \)
13 \( 1 + (0.0581 - 0.998i)T \)
17 \( 1 - T \)
19 \( 1 + (0.939 + 0.342i)T \)
23 \( 1 + (0.939 + 0.342i)T \)
29 \( 1 + (0.0581 - 0.998i)T \)
31 \( 1 + (-0.396 - 0.918i)T \)
41 \( 1 + (0.766 + 0.642i)T \)
43 \( 1 + (-0.766 - 0.642i)T \)
47 \( 1 + (-0.686 - 0.727i)T \)
53 \( 1 + (0.973 + 0.230i)T \)
59 \( 1 + (0.286 - 0.957i)T \)
61 \( 1 + (-0.597 - 0.802i)T \)
67 \( 1 + (-0.286 + 0.957i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (-0.0581 - 0.998i)T \)
79 \( 1 + (0.686 + 0.727i)T \)
83 \( 1 + (-0.993 + 0.116i)T \)
89 \( 1 + (0.0581 - 0.998i)T \)
97 \( 1 + (-0.396 + 0.918i)T \)
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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

−18.60011572679335960953990974562, −17.999329701734463674991987170028, −17.79838963264211930092397451789, −16.46970789644537441409636524314, −15.86630984842700485201952728138, −15.21447625442987998628851752964, −14.66135583350118755363568308045, −14.15155809638232697476428954814, −13.60229158160881983704137663669, −12.99611967512897359744029884341, −12.0523410571207314696862705753, −11.29198901722369509488013435733, −10.62936202857057635214852766708, −9.43748368280234486418316925950, −8.82964200338191304336235401205, −8.32787818666303930101827588468, −7.38134031266147749290090517982, −7.08036898307480095079369321925, −6.305074954177312408429278064, −5.329814981007827693289594780864, −4.642393433273703897881581030102, −3.785577310646876182106141120512, −2.876742614917696569332374615189, −2.54568919876253598145763969055, −1.423169283017967678797986261519, 0.57854860250388314321498586650, 1.53618948404210205378616184460, 2.14608693879140005752693678420, 2.9750791874904070403904797896, 3.86840314688916924908490126252, 4.537373801996904002577012313332, 5.068841350628012509394931953938, 5.68658323381049796942050243072, 7.12863454189923943399064282958, 7.96671719586417910642924343806, 8.38341004500303039226304960722, 9.32150555060982550930233557645, 9.8163115284034775737613451027, 10.54372299148629676013616259970, 11.24488384768380041682526363573, 12.05901870184009444552179901599, 12.94463094777652777549591048607, 13.33591969017573570355307666191, 13.72624448758934007207398302160, 14.76893744509575853091656708166, 15.23889192828241100837468944229, 15.7639186519055228990483996760, 16.81264986150576794591955441761, 17.77013247986634408999615595846, 18.1310079783155408948391464281

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