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Properties

Label 1-4033-4033.563-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.993 + 0.111i$
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 + (−0.396 − 0.918i)3-s + (0.766 + 0.642i)4-s + (0.230 + 0.973i)5-s + (−0.0581 − 0.998i)6-s + (−0.286 + 0.957i)7-s + (0.5 + 0.866i)8-s + (−0.686 + 0.727i)9-s + (−0.116 + 0.993i)10-s + (0.116 + 0.993i)11-s + (0.286 − 0.957i)12-s + (0.286 − 0.957i)13-s + (−0.597 + 0.802i)14-s + (0.802 − 0.597i)15-s + (0.173 + 0.984i)16-s + (0.173 + 0.984i)17-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)2-s + (−0.396 − 0.918i)3-s + (0.766 + 0.642i)4-s + (0.230 + 0.973i)5-s + (−0.0581 − 0.998i)6-s + (−0.286 + 0.957i)7-s + (0.5 + 0.866i)8-s + (−0.686 + 0.727i)9-s + (−0.116 + 0.993i)10-s + (0.116 + 0.993i)11-s + (0.286 − 0.957i)12-s + (0.286 − 0.957i)13-s + (−0.597 + 0.802i)14-s + (0.802 − 0.597i)15-s + (0.173 + 0.984i)16-s + (0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.993 + 0.111i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (563, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.993 + 0.111i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09556546836 + 1.713959892i\)
\(L(\frac12)\) \(\approx\) \(0.09556546836 + 1.713959892i\)
\(L(1)\) \(\approx\) \(1.281008419 + 0.6363541001i\)
\(L(1)\) \(\approx\) \(1.281008419 + 0.6363541001i\)

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 \( 1 + (-0.396 - 0.918i)T \)
5 \( 1 + (0.230 + 0.973i)T \)
7 \( 1 + (-0.286 + 0.957i)T \)
11 \( 1 + (0.116 + 0.993i)T \)
13 \( 1 + (0.286 - 0.957i)T \)
17 \( 1 + (0.173 + 0.984i)T \)
19 \( 1 + (-0.173 - 0.984i)T \)
23 \( 1 + (-0.939 + 0.342i)T \)
29 \( 1 + (-0.918 + 0.396i)T \)
31 \( 1 + (-0.727 - 0.686i)T \)
41 \( 1 + iT \)
43 \( 1 + (0.984 - 0.173i)T \)
47 \( 1 + (-0.549 + 0.835i)T \)
53 \( 1 + (-0.727 + 0.686i)T \)
59 \( 1 + (-0.597 - 0.802i)T \)
61 \( 1 + (-0.448 + 0.893i)T \)
67 \( 1 + (0.230 - 0.973i)T \)
71 \( 1 + (0.939 - 0.342i)T \)
73 \( 1 + (0.993 - 0.116i)T \)
79 \( 1 + (0.286 + 0.957i)T \)
83 \( 1 + (-0.396 - 0.918i)T \)
89 \( 1 + (0.448 - 0.893i)T \)
97 \( 1 + (0.230 + 0.973i)T \)
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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

−18.14655987014837818438311756809, −16.95793250723012921883326365065, −16.59170577172866815376965497427, −16.15888504964581856273754539615, −15.65879693849397417085621624945, −14.3560609994232152910540089471, −14.125578779098058448056787062, −13.48960309418952815544094406596, −12.594617529145404632721284013, −11.97385423246227721477914147125, −11.281709701048537524802454654159, −10.723438322053353032081703105179, −9.88757986282872772923590104426, −9.42629684192958585486045591554, −8.569181059705523999353131196303, −7.53258289779926788816523097830, −6.50055630967355092165965581879, −5.93835127906813786810639180076, −5.25059770624251682404467923270, −4.55099833292570273796550948292, −3.73855983503797850874723094477, −3.60258885284682601572986607191, −2.22142397966321562491242219963, −1.242271692696802673715809943898, −0.32942134808569720869263173794, 1.67248682635565880859731099967, 2.20325234453701966438933699546, 2.92403087098714463664671127300, 3.67635807908467431369043693151, 4.82058656757432566172932520143, 5.63637139811738088732952219562, 6.154081186344681092418701936051, 6.577307006285965713089930835214, 7.674304889369601494660116213832, 7.74111020643959844810770846904, 8.95908722322102387996760735818, 9.95871112905217203854258892019, 10.9624674775742981066129675512, 11.2689026661445735248474694302, 12.24147188283490908320980760582, 12.72476631007060835967285163176, 13.155339776942237490409620577435, 14.04373014185026231360082645096, 14.733444931609442245181213547808, 15.266497156266523717131477324035, 15.780037594285916397412122224713, 16.882235920868844423270743746562, 17.48680886514015476870805496675, 18.07310983071536225890459388489, 18.58255839001430237207902139840

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