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Properties

Label 1-4033-4033.266-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.834 + 0.550i$
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 + (0.893 − 0.448i)3-s + (−0.5 + 0.866i)4-s + (−0.993 + 0.116i)5-s + (−0.835 − 0.549i)6-s + (0.597 + 0.802i)7-s + 8-s + (0.597 − 0.802i)9-s + (0.597 + 0.802i)10-s + (0.597 − 0.802i)11-s + (−0.0581 + 0.998i)12-s + (−0.993 + 0.116i)13-s + (0.396 − 0.918i)14-s + (−0.835 + 0.549i)15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.893 − 0.448i)3-s + (−0.5 + 0.866i)4-s + (−0.993 + 0.116i)5-s + (−0.835 − 0.549i)6-s + (0.597 + 0.802i)7-s + 8-s + (0.597 − 0.802i)9-s + (0.597 + 0.802i)10-s + (0.597 − 0.802i)11-s + (−0.0581 + 0.998i)12-s + (−0.993 + 0.116i)13-s + (0.396 − 0.918i)14-s + (−0.835 + 0.549i)15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.834 + 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.834 + 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.834 + 0.550i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (266, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.834 + 0.550i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1232418003 - 0.4107674582i\)
\(L(\frac12)\) \(\approx\) \(-0.1232418003 - 0.4107674582i\)
\(L(1)\) \(\approx\) \(0.7060478139 - 0.3890814128i\)
\(L(1)\) \(\approx\) \(0.7060478139 - 0.3890814128i\)

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 \( 1 + (0.893 - 0.448i)T \)
5 \( 1 + (-0.993 + 0.116i)T \)
7 \( 1 + (0.597 + 0.802i)T \)
11 \( 1 + (0.597 - 0.802i)T \)
13 \( 1 + (-0.993 + 0.116i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.939 - 0.342i)T \)
23 \( 1 + (0.766 - 0.642i)T \)
29 \( 1 + (0.396 - 0.918i)T \)
31 \( 1 + (-0.286 + 0.957i)T \)
41 \( 1 + (-0.939 + 0.342i)T \)
43 \( 1 + (0.173 - 0.984i)T \)
47 \( 1 + (-0.835 + 0.549i)T \)
53 \( 1 + (-0.835 - 0.549i)T \)
59 \( 1 + (-0.835 + 0.549i)T \)
61 \( 1 + (-0.286 - 0.957i)T \)
67 \( 1 + (-0.0581 - 0.998i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (0.396 + 0.918i)T \)
79 \( 1 + (0.893 + 0.448i)T \)
83 \( 1 + (-0.286 - 0.957i)T \)
89 \( 1 + (0.597 + 0.802i)T \)
97 \( 1 + (-0.286 - 0.957i)T \)
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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

−19.02826953179089387496227907949, −18.11923513471885136274954357245, −17.34266455608525783695564628791, −16.73460620317561081123928960660, −16.19265229534283726826313345757, −15.28519603955476409651102172906, −14.86846657893213211363733784309, −14.531574089033883377281902648074, −13.65438042704687469856985794148, −12.96214338212409923466583504846, −11.979491047565690363045318325765, −11.0247461628381006292738150728, −10.48908139696590655475545377151, −9.59431593554678125452007353785, −9.14065915935391021739032513037, −8.29706420431125370749331475673, −7.70834268442706623288939336757, −7.20508405802691926547502875366, −6.68410597578440329304239222156, −5.11123443113154509183477576603, −4.64641943992303434223237588056, −4.18575772435634541354835910503, −3.243034274806201944670230839319, −2.04567256953844614066385092836, −1.22179828147258238275955539482, 0.13093105837983286405402162917, 1.3070572826542535863138333834, 2.1047921092021272376749848747, 2.80314146185011681715940850941, 3.465333262394551839426644950032, 4.30933326191534553815227754845, 4.875973635753653546859521073003, 6.408193405600461461532645000277, 7.02786431954044686916245660942, 8.0057237713419055976103270209, 8.39792410409714493376562617001, 8.84879102571510483777069009763, 9.56994959554087372384821229986, 10.65640632841742345864687658121, 11.17923543808981721279871300834, 12.012808545117666861040546091858, 12.38862737494341847366001968319, 13.04316099709955529021328127576, 13.9847498194481715037668358219, 14.67559152588356155487248689661, 15.164605325886673286617368873012, 15.97009690452046239641706498966, 17.02147760991284936319270852511, 17.46702722834155067484502446006, 18.4496967351489889006965257124

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