Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $0.928 + 0.371i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  − 2-s + (−0.835 + 0.549i)3-s + 4-s + (−0.597 + 0.802i)5-s + (0.835 − 0.549i)6-s + (0.396 + 0.918i)7-s − 8-s + (0.396 − 0.918i)9-s + (0.597 − 0.802i)10-s + (0.597 + 0.802i)11-s + (−0.835 + 0.549i)12-s + (−0.597 + 0.802i)13-s + (−0.396 − 0.918i)14-s + (0.0581 − 0.998i)15-s + 16-s + (0.5 − 0.866i)17-s + ⋯
L(s,χ)  = 1  − 2-s + (−0.835 + 0.549i)3-s + 4-s + (−0.597 + 0.802i)5-s + (0.835 − 0.549i)6-s + (0.396 + 0.918i)7-s − 8-s + (0.396 − 0.918i)9-s + (0.597 − 0.802i)10-s + (0.597 + 0.802i)11-s + (−0.835 + 0.549i)12-s + (−0.597 + 0.802i)13-s + (−0.396 − 0.918i)14-s + (0.0581 − 0.998i)15-s + 16-s + (0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.928 + 0.371i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.928 + 0.371i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $0.928 + 0.371i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (25, \cdot )$
Sato-Tate  :  $\mu(54)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (0:\ ),\ 0.928 + 0.371i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.5328419087 + 0.1027527945i$
$L(\frac12,\chi)$  $\approx$  $0.5328419087 + 0.1027527945i$
$L(\chi,1)$  $\approx$  0.4645385128 + 0.1924062819i
$L(1,\chi)$  $\approx$  0.4645385128 + 0.1924062819i

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

−18.276043177004871081369072162935, −17.530061647970811464168886085300, −17.21978746068200340177314102656, −16.51305143885561610580724331114, −16.198889863714037401305518460290, −15.22173157677426469280264515101, −14.49504679615018068425631144064, −13.36334395704859256243096193127, −12.81790183127458755461187781065, −12.07653311431662484081860108359, −11.31532350325988139765214987997, −11.07974635267208544239595532610, −10.16983234477863425985774458177, −9.48564814128207649076553910287, −8.43139829587139987750820679145, −7.936776130401295766300155389358, −7.45424706196552592207210311318, −6.63791795345850846588463413271, −5.84629061920123767561843997307, −5.134431290577634853907779109522, −4.179159530702522926641237262021, −3.333067674116148537863633232, −2.09051194191775156540229452998, −1.08341610815833454466266094452, −0.81530000234179214218619075503, 0.35842039148461165845825226774, 1.74690964204744056127364151704, 2.35827149988217128932782378136, 3.435781437095213030511556333788, 4.22945487602415283070673797763, 5.13455287241088403110401748887, 6.03791432402657410027233176734, 6.64193563141921866596318368226, 7.295186694573998453618173505366, 8.03187305952619984085730462646, 8.96632662564243236158607863482, 9.61755526454951193275295409976, 10.18868931043465330458540139474, 10.84325146899361719119390998752, 11.81377114051571099301077965287, 11.92831584826656692573819165981, 12.349772441900364373732716570407, 14.17384410161419555242459007187, 14.76630233182031698180136594888, 15.17936526949828614741426471508, 16.02962675456843167758160087709, 16.411442883943787453272511485002, 17.26947449095403173465870517673, 17.85307453285940552604205504030, 18.48138442684402830815428762736

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