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Properties

Label 1-4033-4033.3087-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.963 - 0.268i$
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 + (−0.0581 − 0.998i)3-s + (−0.5 − 0.866i)4-s + (0.396 + 0.918i)5-s + (0.893 + 0.448i)6-s + (−0.993 − 0.116i)7-s + 8-s + (−0.993 + 0.116i)9-s + (−0.993 − 0.116i)10-s + (−0.993 + 0.116i)11-s + (−0.835 + 0.549i)12-s + (0.396 + 0.918i)13-s + (0.597 − 0.802i)14-s + (0.893 − 0.448i)15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.0581 − 0.998i)3-s + (−0.5 − 0.866i)4-s + (0.396 + 0.918i)5-s + (0.893 + 0.448i)6-s + (−0.993 − 0.116i)7-s + 8-s + (−0.993 + 0.116i)9-s + (−0.993 − 0.116i)10-s + (−0.993 + 0.116i)11-s + (−0.835 + 0.549i)12-s + (0.396 + 0.918i)13-s + (0.597 − 0.802i)14-s + (0.893 − 0.448i)15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.963 - 0.268i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.963 - 0.268i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.963 - 0.268i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (3087, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.963 - 0.268i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03074407196 + 0.2244639840i\)
\(L(\frac12)\) \(\approx\) \(0.03074407196 + 0.2244639840i\)
\(L(1)\) \(\approx\) \(0.5761577458 + 0.1671405707i\)
\(L(1)\) \(\approx\) \(0.5761577458 + 0.1671405707i\)

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 \( 1 + (-0.0581 - 0.998i)T \)
5 \( 1 + (0.396 + 0.918i)T \)
7 \( 1 + (-0.993 - 0.116i)T \)
11 \( 1 + (-0.993 + 0.116i)T \)
13 \( 1 + (0.396 + 0.918i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.939 + 0.342i)T \)
23 \( 1 + (0.766 + 0.642i)T \)
29 \( 1 + (0.597 - 0.802i)T \)
31 \( 1 + (0.973 + 0.230i)T \)
41 \( 1 + (-0.939 - 0.342i)T \)
43 \( 1 + (0.173 + 0.984i)T \)
47 \( 1 + (0.893 - 0.448i)T \)
53 \( 1 + (0.893 + 0.448i)T \)
59 \( 1 + (0.893 - 0.448i)T \)
61 \( 1 + (0.973 - 0.230i)T \)
67 \( 1 + (-0.835 - 0.549i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (0.597 + 0.802i)T \)
79 \( 1 + (-0.0581 + 0.998i)T \)
83 \( 1 + (0.973 - 0.230i)T \)
89 \( 1 + (-0.993 - 0.116i)T \)
97 \( 1 + (0.973 - 0.230i)T \)
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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

−17.96145980228104370716221345217, −17.41044428341619008517310122787, −16.79875758346462000962809063960, −16.18113016669332214394018698683, −15.60818173892729357903073503369, −14.90432088293244747407592168294, −13.574386740498506736492363296963, −13.20673363685925304317698205688, −12.61976219479139231086901631743, −11.95182682118905983162385907642, −10.84445744857162225992623437854, −10.386448006161277780537879712753, −10.045157749252485428886226531301, −9.05648568987968225182216934473, −8.5915169173309395138591423986, −8.20981825787558774565443674335, −6.83552476907839370645115542599, −5.86832590393041931923270367394, −5.17263518237871219120055225384, −4.42003940645393433764676772115, −3.72294712626243629662314388774, −2.818954685700423000886723451185, −2.356198374218837506185918155524, −0.95597058378398137218578704788, −0.0983483779470363317450994169, 1.05531080192764211348758764640, 2.20278425694879098748836383382, 2.68324673154174009808255214617, 3.82366151898120240582034560949, 4.97974153936351352786841893656, 5.8129328972245456240033151349, 6.491177480645558955771959494704, 6.82579753546288868457801240831, 7.44264755621641663531064529740, 8.270986595219269073622392541869, 9.02814373905758031646572865860, 9.76701484296120982192465413433, 10.462584573872492723395080383956, 11.12848995972954885770295560884, 11.98526800926375046593608642432, 13.07358068837195372916805797483, 13.5607994098049907848245679223, 13.87659449796804084503169365912, 14.82855334159862778536033987639, 15.50410640968610419451921547923, 16.13649406662581000678070682093, 16.987672027274895789740409532433, 17.49210479622921954165311373852, 18.24542064372142956501010294816, 18.72849075684875169578517730183

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