Properties

Label 1-4033-4033.2925-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.944 - 0.327i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + (0.0581 + 0.998i)3-s + 4-s + (0.918 − 0.396i)5-s + (−0.0581 − 0.998i)6-s + (−0.993 − 0.116i)7-s − 8-s + (−0.993 + 0.116i)9-s + (−0.918 + 0.396i)10-s + (0.918 + 0.396i)11-s + (0.0581 + 0.998i)12-s + (−0.396 − 0.918i)13-s + (0.993 + 0.116i)14-s + (0.448 + 0.893i)15-s + 16-s + (−0.5 + 0.866i)17-s + ⋯
L(s)  = 1  − 2-s + (0.0581 + 0.998i)3-s + 4-s + (0.918 − 0.396i)5-s + (−0.0581 − 0.998i)6-s + (−0.993 − 0.116i)7-s − 8-s + (−0.993 + 0.116i)9-s + (−0.918 + 0.396i)10-s + (0.918 + 0.396i)11-s + (0.0581 + 0.998i)12-s + (−0.396 − 0.918i)13-s + (0.993 + 0.116i)14-s + (0.448 + 0.893i)15-s + 16-s + (−0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.944 - 0.327i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2925, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.944 - 0.327i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.05929565617 + 0.3515692522i\)
\(L(\frac12)\) \(\approx\) \(-0.05929565617 + 0.3515692522i\)
\(L(1)\) \(\approx\) \(0.6014149653 + 0.2280978190i\)
\(L(1)\) \(\approx\) \(0.6014149653 + 0.2280978190i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + (0.0581 + 0.998i)T \)
5 \( 1 + (0.918 - 0.396i)T \)
7 \( 1 + (-0.993 - 0.116i)T \)
11 \( 1 + (0.918 + 0.396i)T \)
13 \( 1 + (-0.396 - 0.918i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.173 + 0.984i)T \)
23 \( 1 + (0.766 + 0.642i)T \)
29 \( 1 + (-0.116 + 0.993i)T \)
31 \( 1 + (0.727 + 0.686i)T \)
41 \( 1 + (-0.642 - 0.766i)T \)
43 \( 1 + (-0.984 + 0.173i)T \)
47 \( 1 + (-0.998 - 0.0581i)T \)
53 \( 1 + (-0.549 - 0.835i)T \)
59 \( 1 + (-0.893 + 0.448i)T \)
61 \( 1 + (-0.230 - 0.973i)T \)
67 \( 1 + (-0.998 - 0.0581i)T \)
71 \( 1 - T \)
73 \( 1 + (0.993 - 0.116i)T \)
79 \( 1 + (-0.893 + 0.448i)T \)
83 \( 1 + (0.286 - 0.957i)T \)
89 \( 1 + (0.802 + 0.597i)T \)
97 \( 1 + (-0.727 + 0.686i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.228024723577353173339547104708, −17.41954089382272477094383922768, −16.88965032039411682743660915602, −16.51111977103321230008726295381, −15.36977747339272875535453654525, −14.73845742923447648157502634426, −13.830349799013071086543627859506, −13.3870576692018980705312459829, −12.59231109950636922183939392050, −11.57416275080741455722148001006, −11.467008558719982235701271450992, −10.356026266763618786176777228747, −9.487953131057493607065407863276, −9.1800600363337477396865070634, −8.54129525399845508396908103915, −7.40565732709929900850869417683, −6.75025410869613534212740772392, −6.4816400075411000994543120125, −5.90620678989710324384336974620, −4.66564873823195260053374346057, −3.15884345290248913769788530867, −2.714021400582818898355318997857, −2.006001198556013641348898055334, −1.1463552213455189503088812849, −0.14024876002392354216634423883, 1.251048846396935272687955491569, 2.01676042723287748896281774918, 3.13891273205357327073540438415, 3.50950874866714862658664164004, 4.7485488070908229757743740583, 5.550669169782443177484181605769, 6.287900758150487731922296399993, 6.79175149988910424907141922051, 7.94772008560744459513930717391, 8.76860849114487850999097160233, 9.18338429384869723496675985881, 9.93977365841350120470398991389, 10.239981991462249075414215179224, 10.86971899596095675113134596509, 11.9506942790517178331809146473, 12.55411142589176955138254057125, 13.30817373609167777662793018095, 14.3628211969504392403494513391, 14.95029903079568000295397669457, 15.58384176981678481532244669984, 16.3693584224105316926709562431, 16.86827762926285511846539934343, 17.35115364228157413400962407469, 17.86960496547462095218312440876, 18.94165314556919034236540085865

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