Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $-0.248 + 0.968i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.173 − 0.984i)2-s + (0.835 + 0.549i)3-s + (−0.939 − 0.342i)4-s + (0.116 + 0.993i)5-s + (0.686 − 0.727i)6-s + (0.597 + 0.802i)7-s + (−0.5 + 0.866i)8-s + (0.396 + 0.918i)9-s + (0.998 + 0.0581i)10-s + (−0.998 + 0.0581i)11-s + (−0.597 − 0.802i)12-s + (0.597 + 0.802i)13-s + (0.893 − 0.448i)14-s + (−0.448 + 0.893i)15-s + (0.766 + 0.642i)16-s + (−0.766 − 0.642i)17-s + ⋯
L(s,χ)  = 1  + (0.173 − 0.984i)2-s + (0.835 + 0.549i)3-s + (−0.939 − 0.342i)4-s + (0.116 + 0.993i)5-s + (0.686 − 0.727i)6-s + (0.597 + 0.802i)7-s + (−0.5 + 0.866i)8-s + (0.396 + 0.918i)9-s + (0.998 + 0.0581i)10-s + (−0.998 + 0.0581i)11-s + (−0.597 − 0.802i)12-s + (0.597 + 0.802i)13-s + (0.893 − 0.448i)14-s + (−0.448 + 0.893i)15-s + (0.766 + 0.642i)16-s + (−0.766 − 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.248 + 0.968i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.248 + 0.968i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $-0.248 + 0.968i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (532, \cdot )$
Sato-Tate  :  $\mu(108)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (0:\ ),\ -0.248 + 0.968i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.233026249 + 1.589740402i$
$L(\frac12,\chi)$  $\approx$  $1.233026249 + 1.589740402i$
$L(\chi,1)$  $\approx$  1.347990645 + 0.2113206100i
$L(1,\chi)$  $\approx$  1.347990645 + 0.2113206100i

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

−17.98313617772389387817417653298, −17.54145416582070015492429111224, −17.113305761250878366532008794953, −15.93557440700726594091981937772, −15.58168568743864844864128061112, −15.03887756367263847370107105036, −13.89673097051223295712334252922, −13.54494837798544516717538507845, −13.22088220717411506087340998107, −12.487868420353892545990174728981, −11.605972112842823196321537890827, −10.46155322726051274401956015436, −9.74020431486181453548553190497, −8.8486782960828229393003722733, −8.40693042457926576456794011999, −7.721805419300014430862668956467, −7.381279076223104975949539159129, −6.34207236520006610981494579527, −5.52256538490231913530074267198, −4.89288021454001765124152549984, −4.02081003831125910117740772048, −3.44135074091469093386690612898, −2.27127658471034416102669009932, −1.2297680064992277022487694747, −0.470930821226979299848975033617, 1.4781896651266599320657153274, 2.280833419342691007929871423375, 2.703576773949897853439733787093, 3.399725928823650108120981679304, 4.309416594781351450821606318844, 4.93999295707572839925206857661, 5.72729359004087909449401894228, 6.71193692704528661469769738584, 7.86448496712793564003188989598, 8.31762666482566455312302078312, 9.18074505712019525887272821345, 9.71770183549783145752172269207, 10.45090152601655703441728203379, 11.10789772155635302297450738261, 11.53495781615244673576586277543, 12.49316539702318508200006617768, 13.39615060833996685595205814731, 13.866517551494783908972683687123, 14.53196794482009010783094914943, 14.99515119648286461219184073673, 15.76013459177636572299434498153, 16.43538568261392640393144385625, 17.84321270411124458555081037659, 18.18865321961583600905446808750, 18.82790745429122802457671102186

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