Properties

Degree 1
Conductor $ 7 \cdot 29 $
Sign $0.397 + 0.917i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.997 − 0.0747i)2-s + (−0.930 + 0.365i)3-s + (0.988 − 0.149i)4-s + (0.0747 + 0.997i)5-s + (−0.900 + 0.433i)6-s + (0.974 − 0.222i)8-s + (0.733 − 0.680i)9-s + (0.149 + 0.988i)10-s + (−0.680 + 0.733i)11-s + (−0.866 + 0.5i)12-s + (−0.222 + 0.974i)13-s + (−0.433 − 0.900i)15-s + (0.955 − 0.294i)16-s + (−0.866 − 0.5i)17-s + (0.680 − 0.733i)18-s + (0.930 + 0.365i)19-s + ⋯
L(s,χ)  = 1  + (0.997 − 0.0747i)2-s + (−0.930 + 0.365i)3-s + (0.988 − 0.149i)4-s + (0.0747 + 0.997i)5-s + (−0.900 + 0.433i)6-s + (0.974 − 0.222i)8-s + (0.733 − 0.680i)9-s + (0.149 + 0.988i)10-s + (−0.680 + 0.733i)11-s + (−0.866 + 0.5i)12-s + (−0.222 + 0.974i)13-s + (−0.433 − 0.900i)15-s + (0.955 − 0.294i)16-s + (−0.866 − 0.5i)17-s + (0.680 − 0.733i)18-s + (0.930 + 0.365i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.397 + 0.917i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.397 + 0.917i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(203\)    =    \(7 \cdot 29\)
\( \varepsilon \)  =  $0.397 + 0.917i$
motivic weight  =  \(0\)
character  :  $\chi_{203} (19, \cdot )$
Sato-Tate  :  $\mu(84)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 203,\ (0:\ ),\ 0.397 + 0.917i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(1.331532097 + 0.8740329442i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(1.331532097 + 0.8740329442i\)
\(L(\chi,1)\)  \(\approx\)  \(1.366880951 + 0.4367251157i\)
\(L(1,\chi)\)  \(\approx\)  \(1.366880951 + 0.4367251157i\)

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

−26.67331122485256512018187298186, −25.232675388495721675053294584355, −24.289353410976342910398376245296, −24.071672860409116774129112172577, −22.87144333580223022497117694690, −22.14236508676471167019679449749, −21.159193143762002983401157731576, −20.28500215682508463812135016551, −19.18690171370832401270224415060, −17.77449374087346338155610100730, −16.89709595188998830147974371524, −16.03559878652676591371410961780, −15.25711321514554824478903106014, −13.48676142259885576559395388315, −13.118512724655745725199610474338, −12.15349716083025415188489065508, −11.21414767617323850514308453923, −10.234137499068680461597138188861, −8.401988298117905560082589211503, −7.32414836424352914321217118717, −6.00547147670213765525248012968, −5.2898850959965525099909325985, −4.40116932540205042724637922484, −2.697350481314460914772753021390, −1.0591383751074274506821520900, 1.982811996462797791132198919229, 3.35905913097271538714096676099, 4.5919694967929316569335409514, 5.49975222782065957580660284953, 6.7665315803561906784978903241, 7.26111655770274189961466430347, 9.610832246869650602731730747151, 10.56757034363232588211549361450, 11.39417203225524965332540019694, 12.18250552023983579246997041376, 13.41443111026459295456444554727, 14.44407458531772853673660733686, 15.45821658359763216066643899032, 16.08405990222253527753935881657, 17.39706822507952227911695129982, 18.32663454904077279967524784136, 19.49047341697487933444501496157, 20.8537010619968282866864518773, 21.51882915828347021266150290159, 22.50487635794515931704282591295, 22.96519493185997867751072849369, 23.84726186258425111268747576710, 24.90459208672531651914442523298, 26.159363239879686501560265510448, 26.87604444280840992033184522665

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