Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $0.866 + 0.498i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.642 + 0.766i)2-s + (0.973 − 0.230i)3-s + (−0.173 + 0.984i)4-s + (−0.835 + 0.549i)5-s + (0.802 + 0.597i)6-s + (−0.0581 − 0.998i)7-s + (−0.866 + 0.5i)8-s + (0.893 − 0.448i)9-s + (−0.957 − 0.286i)10-s + (−0.957 + 0.286i)11-s + (0.0581 + 0.998i)12-s + (−0.998 + 0.0581i)13-s + (0.727 − 0.686i)14-s + (−0.686 + 0.727i)15-s + (−0.939 − 0.342i)16-s + (0.342 − 0.939i)17-s + ⋯
L(s,χ)  = 1  + (0.642 + 0.766i)2-s + (0.973 − 0.230i)3-s + (−0.173 + 0.984i)4-s + (−0.835 + 0.549i)5-s + (0.802 + 0.597i)6-s + (−0.0581 − 0.998i)7-s + (−0.866 + 0.5i)8-s + (0.893 − 0.448i)9-s + (−0.957 − 0.286i)10-s + (−0.957 + 0.286i)11-s + (0.0581 + 0.998i)12-s + (−0.998 + 0.0581i)13-s + (0.727 − 0.686i)14-s + (−0.686 + 0.727i)15-s + (−0.939 − 0.342i)16-s + (0.342 − 0.939i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.866 + 0.498i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.866 + 0.498i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $0.866 + 0.498i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (2764, \cdot )$
Sato-Tate  :  $\mu(108)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (1:\ ),\ 0.866 + 0.498i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.728678977 + 0.4620108743i$
$L(\frac12,\chi)$  $\approx$  $1.728678977 + 0.4620108743i$
$L(\chi,1)$  $\approx$  1.179166624 + 0.5315269695i
$L(1,\chi)$  $\approx$  1.179166624 + 0.5315269695i

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

−18.70336727314843295046595499321, −17.82238405659802865128242064070, −16.65180960119461025997677504932, −15.76541777017502275609376762929, −15.44301907425231387531529533188, −14.76167455223515518546988359714, −14.313508125084622096244549208545, −13.13626004351735259058110185411, −12.80876793103300530281947087606, −12.33040054763846925331927953531, −11.42968778718901102396093150282, −10.72125707904952069307510874458, −9.970102974983058375319052380353, −9.17833644114147241093022736393, −8.68291724906759907333289513218, −7.92455747546188044364646356897, −7.191585818196028052841024251028, −5.93763948414815208107513735155, −5.23594963968475772722873700251, −4.55965700945882522281641437021, −3.87869060540856727796222804835, −3.0899564238541364085254866584, −2.378268614775279374804406428249, −1.85615017801034438216401353752, −0.48296209954777090198269806995, 0.29903836406234445739106062461, 1.81546945995835012920672356774, 2.80381366901901940434720638767, 3.34285315102398980024440256063, 4.06599541310030718068217391918, 4.659081997934345850175397339043, 5.612954570201298529061067127394, 6.730979141865311224351152918044, 7.388513820435970418925330041810, 7.61860305347283328586003349744, 8.1316456142973097800606515610, 9.2512628540782285643989198649, 9.956766051521623755781538707652, 10.75206813248782053220506322178, 11.75695239096683503680932749101, 12.427998595750148800110993351095, 13.038905635986326102570556995934, 13.78317885272244004251187547661, 14.35930400179387347702813743192, 14.84884937535056909007889377557, 15.442867900065968249888287468769, 16.26849367296096457393326005217, 16.614252247057316867634169956696, 17.825418701005094709213242819575, 18.23643131843134089258036051680

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