Properties

Degree 1
Conductor $ 13 \cdot 89 $
Sign $0.202 + 0.979i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.945 + 0.327i)2-s + (−0.992 − 0.118i)3-s + (0.786 + 0.618i)4-s + (−0.841 + 0.540i)5-s + (−0.899 − 0.436i)6-s + (−0.739 − 0.672i)7-s + (0.540 + 0.841i)8-s + (0.971 + 0.235i)9-s + (−0.971 + 0.235i)10-s + (−0.998 − 0.0475i)11-s + (−0.707 − 0.707i)12-s + (−0.479 − 0.877i)14-s + (0.899 − 0.436i)15-s + (0.235 + 0.971i)16-s + (0.945 − 0.327i)17-s + (0.841 + 0.540i)18-s + ⋯
L(s,χ)  = 1  + (0.945 + 0.327i)2-s + (−0.992 − 0.118i)3-s + (0.786 + 0.618i)4-s + (−0.841 + 0.540i)5-s + (−0.899 − 0.436i)6-s + (−0.739 − 0.672i)7-s + (0.540 + 0.841i)8-s + (0.971 + 0.235i)9-s + (−0.971 + 0.235i)10-s + (−0.998 − 0.0475i)11-s + (−0.707 − 0.707i)12-s + (−0.479 − 0.877i)14-s + (0.899 − 0.436i)15-s + (0.235 + 0.971i)16-s + (0.945 − 0.327i)17-s + (0.841 + 0.540i)18-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.202 + 0.979i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.202 + 0.979i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(1157\)    =    \(13 \cdot 89\)
\( \varepsilon \)  =  $0.202 + 0.979i$
motivic weight  =  \(0\)
character  :  $\chi_{1157} (58, \cdot )$
Sato-Tate  :  $\mu(264)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 1157,\ (0:\ ),\ 0.202 + 0.979i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.058826109 + 0.8626787433i$
$L(\frac12,\chi)$  $\approx$  $1.058826109 + 0.8626787433i$
$L(\chi,1)$  $\approx$  1.060382902 + 0.3281002111i
$L(1,\chi)$  $\approx$  1.060382902 + 0.3281002111i

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

−21.232039981212211123794084032296, −20.60416086417323722837258033865, −19.58722388261036224564078437597, −18.91396204851552759508961250884, −18.249222850360062977237374487673, −16.938043304229475482015294453345, −16.19604278429558720174367450102, −15.661746672224080065447285445895, −15.236752315936518838547529431281, −13.93025963756153317868351461474, −12.90048693417828753393239434236, −12.4775259210888881530271993889, −11.84799665060010949625718309, −11.18484778729803421592592245282, −10.14864187973291535872223048927, −9.60502265538406108086500377562, −8.09898713892463596578531583618, −7.26386852330506840300823476895, −6.28029142688038174250655078734, −5.3199791333905926013941018706, −5.11916685019288647025764174383, −3.79525424242472350809823581855, −3.24655180293838779866580239740, −1.83645556001386922124948680701, −0.58358094122315325465707413043, 0.95430948072527179934408867882, 2.77246067701670815139487230066, 3.34901961501024115907842649888, 4.50875403076738504531118072861, 5.04510184406324182283732958658, 6.191817407467159312433208288710, 6.81743037229830291874587007527, 7.47715499983572141332961135271, 8.214080778817705837648018430301, 10.096473687408083541046777100185, 10.48712651431238283687895522409, 11.47479733687913400140326744840, 12.04740947143120848715909495738, 12.81808739909494279675601266536, 13.55695403814368723767499116654, 14.396203651112319440238608166383, 15.49451854192617418411133944979, 16.00760189358199232583471525130, 16.461124603674309220656099337276, 17.41611274850887567673089651647, 18.352940953662762924437831389246, 19.10159071324433301796265326792, 20.03961561269540925619913862001, 20.821905692870331341537920281474, 21.751439076707874305469403445627

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