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Properties

Label 1-4033-4033.1026-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.966 - 0.258i$
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 + 3-s + (−0.5 + 0.866i)4-s − 5-s + (0.5 + 0.866i)6-s + 7-s − 8-s + 9-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (−0.5 + 0.866i)12-s − 13-s + (0.5 + 0.866i)14-s − 15-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + 3-s + (−0.5 + 0.866i)4-s − 5-s + (0.5 + 0.866i)6-s + 7-s − 8-s + 9-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (−0.5 + 0.866i)12-s − 13-s + (0.5 + 0.866i)14-s − 15-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.966 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.966 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.966 - 0.258i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1026, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.966 - 0.258i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2344106996 + 1.783477542i\)
\(L(\frac12)\) \(\approx\) \(-0.2344106996 + 1.783477542i\)
\(L(1)\) \(\approx\) \(1.083274928 + 0.9285580541i\)
\(L(1)\) \(\approx\) \(1.083274928 + 0.9285580541i\)

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

−18.41886349287692665033510556422, −17.80457369105458399118882772866, −16.51226627954493130747715307778, −15.84077296780384883814848457273, −15.11515508865017971052463341871, −14.577681404820744147689934191232, −13.917036067959282272476987391753, −13.57826987827587196619685697535, −12.381133822479941107114935041273, −11.9849751024705106743982167034, −11.46195225189112426679693373950, −10.37755171936605636025998989071, −10.07825429069116346228077108788, −9.03592784598842115202214046743, −8.29790399809042192902486626601, −7.86056232325833723314290183231, −7.10805909629457152783756466468, −5.86920914013898004384150077353, −4.837221335511266988462415540366, −4.59585747864652519730251282924, −3.461240400327381657013443279250, −3.16587416557300189880339112526, −2.21943655263246725925594880837, −1.443510711230486274667723555548, −0.35323061138685853942421875367, 1.366300720955424583676482217546, 2.47190332640152726037355974440, 3.13191496894794255929095529783, 4.050921918230193196269399248112, 4.70185828848972007683518294571, 5.00371515932131302143626019229, 6.34984800768877230990634910331, 7.25101029487001388804564240504, 7.74761353216688026464629810573, 8.03683817394516963387037566831, 8.81754005231109667032835574391, 9.66307825431775313027163798515, 10.44955367043809241105465451738, 11.572173437805317327985014005638, 12.220701666039358768887117177225, 12.73101085733410971988627360130, 13.5586664818003772654447009133, 14.41513362461366503750967326815, 14.70342962423466226752735338440, 15.41782871223048292190977166549, 15.6851018625825318616413287276, 16.70518479880121096236344099580, 17.38347196067864875236012162076, 18.21472203139118263776989753084, 18.62369980421418805972925375248

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