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Properties

Label 1-4033-4033.3841-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.201 + 0.979i$
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 + (0.396 − 0.918i)3-s + (−0.5 − 0.866i)4-s + (0.286 + 0.957i)5-s + (−0.597 − 0.802i)6-s + (−0.686 + 0.727i)7-s − 8-s + (−0.686 − 0.727i)9-s + (0.973 + 0.230i)10-s + (0.973 − 0.230i)11-s + (−0.993 + 0.116i)12-s + (0.286 + 0.957i)13-s + (0.286 + 0.957i)14-s + (0.993 + 0.116i)15-s + (−0.5 + 0.866i)16-s − 17-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (0.396 − 0.918i)3-s + (−0.5 − 0.866i)4-s + (0.286 + 0.957i)5-s + (−0.597 − 0.802i)6-s + (−0.686 + 0.727i)7-s − 8-s + (−0.686 − 0.727i)9-s + (0.973 + 0.230i)10-s + (0.973 − 0.230i)11-s + (−0.993 + 0.116i)12-s + (0.286 + 0.957i)13-s + (0.286 + 0.957i)14-s + (0.993 + 0.116i)15-s + (−0.5 + 0.866i)16-s − 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.201 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.201 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.201 + 0.979i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (3841, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.201 + 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1482933527 - 0.1819077473i\)
\(L(\frac12)\) \(\approx\) \(-0.1482933527 - 0.1819077473i\)
\(L(1)\) \(\approx\) \(0.9080185004 - 0.6461532358i\)
\(L(1)\) \(\approx\) \(0.9080185004 - 0.6461532358i\)

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 \( 1 + (0.396 - 0.918i)T \)
5 \( 1 + (0.286 + 0.957i)T \)
7 \( 1 + (-0.686 + 0.727i)T \)
11 \( 1 + (0.973 - 0.230i)T \)
13 \( 1 + (0.286 + 0.957i)T \)
17 \( 1 - T \)
19 \( 1 + (-0.173 - 0.984i)T \)
23 \( 1 + (-0.173 - 0.984i)T \)
29 \( 1 + (0.286 + 0.957i)T \)
31 \( 1 + (-0.893 - 0.448i)T \)
41 \( 1 + (-0.939 + 0.342i)T \)
43 \( 1 + (0.939 - 0.342i)T \)
47 \( 1 + (0.597 - 0.802i)T \)
53 \( 1 + (0.396 - 0.918i)T \)
59 \( 1 + (0.993 + 0.116i)T \)
61 \( 1 + (0.0581 - 0.998i)T \)
67 \( 1 + (-0.993 - 0.116i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (-0.286 + 0.957i)T \)
79 \( 1 + (-0.597 + 0.802i)T \)
83 \( 1 + (-0.835 - 0.549i)T \)
89 \( 1 + (0.286 + 0.957i)T \)
97 \( 1 + (-0.893 + 0.448i)T \)
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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

−19.15963927047537809201701671489, −17.72390044499980159278586168609, −17.40451947381570594221701928461, −16.71291234206160996786430515761, −16.18830838645125444363461776044, −15.63414634029216263458558604798, −15.00835521666374231517042970723, −14.15844465470474135827746649385, −13.61531288607289307746367628376, −13.083642489288824643216674857908, −12.34600208648682479727244608425, −11.52676822448324106577541967210, −10.45601631908123107351264450182, −9.77975274740014380894539048945, −9.123923465783881300800448318679, −8.61722779636107264000935410957, −7.815660447007871117975837108089, −7.089820865515074762366267537573, −5.917146239357975439054187578345, −5.7401869091702628662045615982, −4.59924545387120911174630556025, −4.09712601831215483936554091577, −3.614175409616923552588621922534, −2.654986048934735060616757305387, −1.36333323860946847811057725068, 0.0487554532262347067456426999, 1.36518567402452229708994774401, 2.26509235332485415652809521506, 2.52664814367964486755380319375, 3.48825356946355851909618046325, 4.04831568052747958114109140639, 5.277984345314527916985180128914, 6.221262309545057576687648321276, 6.60310655361272270352626680480, 7.051652630781415050359805788859, 8.675653015456969868464401355709, 8.92288270404689213725782683240, 9.59178403489805919431710904897, 10.57606304044040278590397104160, 11.383190547735524477918939421, 11.72580341007378087588541992517, 12.56718877849467989413807007566, 13.20835494368017363372666344759, 13.77335405170293164002966685387, 14.4270100821279260833299462101, 14.8977854792329235119419184757, 15.65166116737262772510686289594, 16.73766262249210288013035527478, 17.73643520267675212661483291397, 18.24445104060456104448042608839

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