Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $-0.944 - 0.327i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $-0.944 - 0.327i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (2925, \cdot )$
Sato-Tate  :  $\mu(108)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (0:\ ),\ -0.944 - 0.327i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.05929565617 + 0.3515692522i$
$L(\frac12,\chi)$  $\approx$  $-0.05929565617 + 0.3515692522i$
$L(\chi,1)$  $\approx$  0.6014149653 + 0.2280978190i
$L(1,\chi)$  $\approx$  0.6014149653 + 0.2280978190i

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

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