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  :  $\chi_{4033} (232, \cdot )$
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

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

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