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Properties

Label 1-4011-4011.23-r1-0-0
Degree $1$
Conductor $4011$
Sign $0.345 - 0.938i$
Analytic cond. $431.041$
Root an. cond. $431.041$
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  + (0.319 + 0.947i)2-s + (−0.795 + 0.605i)4-s + (−0.904 + 0.426i)5-s + (−0.828 − 0.560i)8-s + (−0.693 − 0.720i)10-s + (0.191 − 0.981i)11-s + (0.997 − 0.0660i)13-s + (0.266 − 0.963i)16-s + (0.949 − 0.314i)17-s + (−0.693 + 0.720i)19-s + (0.461 − 0.887i)20-s + (0.991 − 0.131i)22-s + (−0.471 − 0.882i)23-s + (0.635 − 0.771i)25-s + (0.381 + 0.924i)26-s + ⋯
L(s)  = 1  + (0.319 + 0.947i)2-s + (−0.795 + 0.605i)4-s + (−0.904 + 0.426i)5-s + (−0.828 − 0.560i)8-s + (−0.693 − 0.720i)10-s + (0.191 − 0.981i)11-s + (0.997 − 0.0660i)13-s + (0.266 − 0.963i)16-s + (0.949 − 0.314i)17-s + (−0.693 + 0.720i)19-s + (0.461 − 0.887i)20-s + (0.991 − 0.131i)22-s + (−0.471 − 0.882i)23-s + (0.635 − 0.771i)25-s + (0.381 + 0.924i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.345 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.345 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4011\)    =    \(3 \cdot 7 \cdot 191\)
Sign: $0.345 - 0.938i$
Analytic conductor: \(431.041\)
Root analytic conductor: \(431.041\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4011} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4011,\ (1:\ ),\ 0.345 - 0.938i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9377979233 - 0.6542300625i\)
\(L(\frac12)\) \(\approx\) \(0.9377979233 - 0.6542300625i\)
\(L(1)\) \(\approx\) \(0.9045383936 + 0.3414798876i\)
\(L(1)\) \(\approx\) \(0.9045383936 + 0.3414798876i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
191 \( 1 \)
good2 \( 1 + (0.319 + 0.947i)T \)
5 \( 1 + (-0.904 + 0.426i)T \)
11 \( 1 + (0.191 - 0.981i)T \)
13 \( 1 + (0.997 - 0.0660i)T \)
17 \( 1 + (0.949 - 0.314i)T \)
19 \( 1 + (-0.693 + 0.720i)T \)
23 \( 1 + (-0.471 - 0.882i)T \)
29 \( 1 + (0.148 - 0.988i)T \)
31 \( 1 + (0.0275 - 0.999i)T \)
37 \( 1 + (0.0275 + 0.999i)T \)
41 \( 1 + (0.677 - 0.735i)T \)
43 \( 1 + (-0.627 - 0.778i)T \)
47 \( 1 + (0.857 - 0.514i)T \)
53 \( 1 + (0.421 + 0.906i)T \)
59 \( 1 + (-0.565 - 0.824i)T \)
61 \( 1 + (0.952 - 0.303i)T \)
67 \( 1 + (0.411 - 0.911i)T \)
71 \( 1 + (-0.490 - 0.871i)T \)
73 \( 1 + (0.391 + 0.920i)T \)
79 \( 1 + (-0.782 + 0.622i)T \)
83 \( 1 + (0.999 - 0.0330i)T \)
89 \( 1 + (0.709 + 0.705i)T \)
97 \( 1 + (0.180 - 0.983i)T \)
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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

−18.60662604669473658377828694425, −17.85280306354601834043364376286, −17.32496155614182170273107765285, −16.19134180137881771279712230988, −15.77514036785368186787335110835, −14.7074728137737118027460801758, −14.55610538478037296939606984642, −13.32431642060638782612688676526, −12.9224208251994336621089261117, −12.16324189444084697610426464491, −11.73111217992014476418767695850, −10.90353389552110241366959793680, −10.3842700424797417766602433713, −9.41870956910630219596517778986, −8.86677709027547721438027959197, −8.13909220115099459371755425425, −7.319506322567608224024510417585, −6.37843038674209313170682388962, −5.42999924793208535251433262722, −4.7640696163615908982596727588, −3.98209765609758732004756895949, −3.539738809000048471159559434238, −2.58174245460586197888495854260, −1.489149350793960625195399631118, −1.01534254441852493516483979011, 0.20793101528826178945709694856, 0.87447332621821706039611963692, 2.45260571767973579669250747358, 3.46228919354329672822879273416, 3.828846270871891548352554863759, 4.57674213210936262923779072626, 5.71915777087073827865525087313, 6.13616828143582766936252065226, 6.86649735787791249384584088281, 7.78728218980146150100536279910, 8.2462362138952459361516707927, 8.740101469152034769087599881846, 9.8180362053490316305477618764, 10.61730316818522648849347133224, 11.43083773899576640107513551924, 12.10014427555545740670799225712, 12.71487339267307397769875001856, 13.84953385623222423197532910691, 13.93776190580062076618597324767, 14.96714705183307816941556909288, 15.38393687723366899004722680185, 16.16938511141852053284083162975, 16.619986061013793541885870250885, 17.230659368926628619312695340299, 18.38151941761287136754084204712

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