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Properties

Label 1-4033-4033.251-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.821 + 0.570i$
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 + (0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + (−0.866 + 0.5i)5-s + 6-s + (−0.5 − 0.866i)7-s − 8-s + (−0.5 + 0.866i)9-s + i·10-s + i·11-s + (0.5 − 0.866i)12-s + (0.5 + 0.866i)13-s − 14-s + (−0.866 − 0.5i)15-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + (−0.866 + 0.5i)5-s + 6-s + (−0.5 − 0.866i)7-s − 8-s + (−0.5 + 0.866i)9-s + i·10-s + i·11-s + (0.5 − 0.866i)12-s + (0.5 + 0.866i)13-s − 14-s + (−0.866 − 0.5i)15-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.821 + 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.821 + 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.821 + 0.570i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.821 + 0.570i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2137912331 + 0.6820027724i\)
\(L(\frac12)\) \(\approx\) \(0.2137912331 + 0.6820027724i\)
\(L(1)\) \(\approx\) \(1.029511910 + 0.01982766621i\)
\(L(1)\) \(\approx\) \(1.029511910 + 0.01982766621i\)

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 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + iT \)
13 \( 1 + (0.5 + 0.866i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + T \)
29 \( 1 - iT \)
31 \( 1 - iT \)
41 \( 1 + (-0.866 + 0.5i)T \)
43 \( 1 + iT \)
47 \( 1 + iT \)
53 \( 1 + (-0.866 - 0.5i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (-0.866 + 0.5i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 - T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 - iT \)
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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

−18.262364652881424261663733755, −17.54732468834487650687301725931, −16.63031332633737804897068061625, −15.97259678078058945368853747148, −15.44442955107488786868163317900, −14.99746610335346392409345950371, −13.95870713688637563283807303876, −13.42901163837775091303036613628, −12.88606428285575690558844108576, −12.22820347103868786286424863834, −11.694807511081114236891240036037, −10.88816408521760445051423395448, −9.297739442867132874284771072845, −8.72822631541257712352080033930, −8.57143828960976023470952619355, −7.60013570311173975486985254686, −6.976031523703737529298542643953, −6.384960133118492176442229924317, −5.3268969197476628900195137271, −5.085380527430117212044259649410, −3.62697516273087408452639462138, −3.22035488417555295592812128384, −2.62004869319198670452582782623, −1.06228055525646714281569349598, −0.17547047121730111323528532314, 1.34473653004784240799063171768, 2.268299287626413519519549060785, 3.18115105978006322534598416067, 3.73398499494744460721996476383, 4.33041326702980829126413131331, 4.71443245788535876712431520460, 6.02217338096276222680373137282, 6.72272451959937969746053103799, 7.68488238684085069722918934617, 8.39968600653398420018172544505, 9.42604791661099373727985418079, 9.842737756890658021617899537228, 10.51625047929468892773516886282, 11.21335897183698731908675018561, 11.60507643629195280545877956734, 12.705629864331646830954549266497, 13.21600687143478590295271262385, 14.11767064868389363547985683539, 14.53335738771985533344648935298, 15.29363149587171551426842799956, 15.70754694120461705010940789650, 16.62445890349135915697383909395, 17.305180152381808651607966267847, 18.38771747459105228971638432758, 19.13688587021554080525978082277

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