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Properties

Label 1-4033-4033.3431-r1-0-0
Degree $1$
Conductor $4033$
Sign $0.321 + 0.947i$
Analytic cond. $433.406$
Root an. cond. $433.406$
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  + (−0.342 − 0.939i)2-s + (−0.993 + 0.116i)3-s + (−0.766 + 0.642i)4-s + (0.286 + 0.957i)5-s + (0.448 + 0.893i)6-s + (−0.686 + 0.727i)7-s + (0.866 + 0.5i)8-s + (0.973 − 0.230i)9-s + (0.802 − 0.597i)10-s + (0.802 + 0.597i)11-s + (0.686 − 0.727i)12-s + (0.727 + 0.686i)13-s + (0.918 + 0.396i)14-s + (−0.396 − 0.918i)15-s + (0.173 − 0.984i)16-s + (0.984 + 0.173i)17-s + ⋯
L(s)  = 1  + (−0.342 − 0.939i)2-s + (−0.993 + 0.116i)3-s + (−0.766 + 0.642i)4-s + (0.286 + 0.957i)5-s + (0.448 + 0.893i)6-s + (−0.686 + 0.727i)7-s + (0.866 + 0.5i)8-s + (0.973 − 0.230i)9-s + (0.802 − 0.597i)10-s + (0.802 + 0.597i)11-s + (0.686 − 0.727i)12-s + (0.727 + 0.686i)13-s + (0.918 + 0.396i)14-s + (−0.396 − 0.918i)15-s + (0.173 − 0.984i)16-s + (0.984 + 0.173i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.321 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.321 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.321 + 0.947i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (3431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ 0.321 + 0.947i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8520576919 + 0.6107740501i\)
\(L(\frac12)\) \(\approx\) \(0.8520576919 + 0.6107740501i\)
\(L(1)\) \(\approx\) \(0.6679183837 + 0.007360396174i\)
\(L(1)\) \(\approx\) \(0.6679183837 + 0.007360396174i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.342 - 0.939i)T \)
3 \( 1 + (-0.993 + 0.116i)T \)
5 \( 1 + (0.286 + 0.957i)T \)
7 \( 1 + (-0.686 + 0.727i)T \)
11 \( 1 + (0.802 + 0.597i)T \)
13 \( 1 + (0.727 + 0.686i)T \)
17 \( 1 + (0.984 + 0.173i)T \)
19 \( 1 + (-0.984 - 0.173i)T \)
23 \( 1 + (0.342 - 0.939i)T \)
29 \( 1 + (-0.993 + 0.116i)T \)
31 \( 1 + (0.973 - 0.230i)T \)
41 \( 1 - iT \)
43 \( 1 + (0.173 - 0.984i)T \)
47 \( 1 + (-0.998 + 0.0581i)T \)
53 \( 1 + (0.230 - 0.973i)T \)
59 \( 1 + (0.918 - 0.396i)T \)
61 \( 1 + (-0.835 - 0.549i)T \)
67 \( 1 + (-0.957 - 0.286i)T \)
71 \( 1 + (0.939 + 0.342i)T \)
73 \( 1 + (0.597 - 0.802i)T \)
79 \( 1 + (-0.727 + 0.686i)T \)
83 \( 1 + (0.993 - 0.116i)T \)
89 \( 1 + (0.835 + 0.549i)T \)
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

−17.83978109901906780351723659627, −17.28778279596869849441315178114, −16.68217867912215349826159391273, −16.46397790313346528295073532718, −15.77187507735844857845987957902, −14.96734392955468721631324698253, −13.98352303402363442221859546988, −13.20296161682528439861466377022, −13.046456335326362283562069707715, −12.065874994441008523900020083446, −11.18134127924073201970618809220, −10.403582569323951451938554626228, −9.773356989639515626040701710176, −9.19758172950057692497748907150, −8.26388955844159797479791125980, −7.65624009934235293913015343578, −6.783859927439670199309783450888, −5.94820194977762092955358761957, −5.87084756687927067661451630825, −4.82610160385090427079155769940, −4.15713756324182803129540655041, −3.37748495958339083230272056110, −1.44826456919896415848930536897, −1.06687471391940702332974192392, −0.34243173183990997206936940112, 0.65288473680024213560209459655, 1.77734169135634060947047575735, 2.27666140749753394893656703556, 3.46624959621482601989068890537, 3.890189785561371527483867053595, 4.86470621710167898486627844062, 5.78759599963351462927383428533, 6.5436457161774837250201891566, 6.94906797679965320824122578725, 8.09498726984184747007145271574, 9.118150314920408752845730283203, 9.54829041204924527362045452581, 10.35063306499293764847928930400, 10.75988792718404444094541915308, 11.631502218621853550700193988703, 12.020329872414103757095818431810, 12.74435310047613996232689481885, 13.36733909305285861050797865080, 14.35930929352797806717294529346, 14.980795647167985724841113865908, 15.84502255464187593710436742007, 16.78370839868316985684602638646, 17.06678180757415716821375780193, 17.95151719666485909435698654930, 18.50529111670909868379381653834

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