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Properties

Label 1-4033-4033.836-r1-0-0
Degree $1$
Conductor $4033$
Sign $-0.971 + 0.238i$
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  + (−0.866 + 0.5i)2-s + (−0.597 + 0.802i)3-s + (0.5 − 0.866i)4-s + (0.230 + 0.973i)5-s + (0.116 − 0.993i)6-s + (−0.286 + 0.957i)7-s + i·8-s + (−0.286 − 0.957i)9-s + (−0.686 − 0.727i)10-s + (0.686 − 0.727i)11-s + (0.396 + 0.918i)12-s + (0.230 + 0.973i)13-s + (−0.230 − 0.973i)14-s + (−0.918 − 0.396i)15-s + (−0.5 − 0.866i)16-s + i·17-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (−0.597 + 0.802i)3-s + (0.5 − 0.866i)4-s + (0.230 + 0.973i)5-s + (0.116 − 0.993i)6-s + (−0.286 + 0.957i)7-s + i·8-s + (−0.286 − 0.957i)9-s + (−0.686 − 0.727i)10-s + (0.686 − 0.727i)11-s + (0.396 + 0.918i)12-s + (0.230 + 0.973i)13-s + (−0.230 − 0.973i)14-s + (−0.918 − 0.396i)15-s + (−0.5 − 0.866i)16-s + i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.971 + 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.971 + 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.971 + 0.238i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (836, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ -0.971 + 0.238i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1790414491 + 1.479779872i\)
\(L(\frac12)\) \(\approx\) \(0.1790414491 + 1.479779872i\)
\(L(1)\) \(\approx\) \(0.5176492031 + 0.5035674793i\)
\(L(1)\) \(\approx\) \(0.5176492031 + 0.5035674793i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (-0.597 + 0.802i)T \)
5 \( 1 + (0.230 + 0.973i)T \)
7 \( 1 + (-0.286 + 0.957i)T \)
11 \( 1 + (0.686 - 0.727i)T \)
13 \( 1 + (0.230 + 0.973i)T \)
17 \( 1 + iT \)
19 \( 1 + (0.984 + 0.173i)T \)
23 \( 1 + (0.984 + 0.173i)T \)
29 \( 1 + (-0.230 - 0.973i)T \)
31 \( 1 + (0.998 + 0.0581i)T \)
41 \( 1 + (0.939 + 0.342i)T \)
43 \( 1 + (0.342 - 0.939i)T \)
47 \( 1 + (-0.993 + 0.116i)T \)
53 \( 1 + (0.597 - 0.802i)T \)
59 \( 1 + (0.918 + 0.396i)T \)
61 \( 1 + (0.549 + 0.835i)T \)
67 \( 1 + (-0.396 + 0.918i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (-0.973 - 0.230i)T \)
79 \( 1 + (-0.116 - 0.993i)T \)
83 \( 1 + (0.893 + 0.448i)T \)
89 \( 1 + (-0.230 - 0.973i)T \)
97 \( 1 + (0.998 - 0.0581i)T \)
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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

−17.764949867069050004300973782628, −17.535582381215083978329189722439, −16.82473860747976661719244951435, −16.27177279717106904822304919843, −15.729486598638112970264289637926, −14.40201681561244545829780708763, −13.46080355792981362209520357968, −13.0680030201332013515826914694, −12.42395565894291994466247528953, −11.82028054218838460576655257193, −11.12012236855907776015149201835, −10.39805414546505796932010761945, −9.631831455329813581664692692821, −9.10328249942510464258739740378, −8.12798822829048062163984801202, −7.51622970126821765753521319660, −6.97446148045565555178116414785, −6.22547435951625733378432225733, −5.14303141569435855847697133077, −4.53697580207721599929684451989, −3.41909319708499019825886938937, −2.590536237776065273125874628003, −1.500580365547557483060261522, −0.906879054860879989206559784, −0.55335409316420166007371551658, 0.72148333885666127443053379943, 1.72716741225435665880060441895, 2.73994326626030892444690889099, 3.49550377901026385907860881083, 4.46723455274194464018258404294, 5.71072001300761838958454371753, 5.86863760353329917514328454099, 6.59327824130656829066367955360, 7.22781916825059627915092561654, 8.4775973440018555642293201257, 8.9158804347145322650209563581, 9.72450690855959776430959922419, 10.08381189825058971911955372997, 11.0488095475508636334828371404, 11.53335177993150596725191350584, 11.91419735605190540356316862892, 13.31834934750318802257626667539, 14.23270867758357008769215936506, 14.81601400694043335510297150120, 15.25709384160905143906975127771, 16.11606665646560056142506399475, 16.42382257187613768397388715426, 17.41325296012169921575056786609, 17.69548482690468169357376762515, 18.61941380570482527244646809100

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