Properties

Degree 1
Conductor 151
Sign $0.912 - 0.409i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.104 − 0.994i)2-s + (−0.425 + 0.904i)3-s + (−0.978 + 0.207i)4-s + (0.985 − 0.166i)5-s + (0.944 + 0.328i)6-s + (−0.895 − 0.444i)7-s + (0.309 + 0.951i)8-s + (−0.637 − 0.770i)9-s + (−0.268 − 0.963i)10-s + (0.996 + 0.0836i)11-s + (0.228 − 0.973i)12-s + (0.387 + 0.921i)13-s + (−0.348 + 0.937i)14-s + (−0.268 + 0.963i)15-s + (0.913 − 0.406i)16-s + (0.832 + 0.553i)17-s + ⋯
L(s,χ)  = 1  + (−0.104 − 0.994i)2-s + (−0.425 + 0.904i)3-s + (−0.978 + 0.207i)4-s + (0.985 − 0.166i)5-s + (0.944 + 0.328i)6-s + (−0.895 − 0.444i)7-s + (0.309 + 0.951i)8-s + (−0.637 − 0.770i)9-s + (−0.268 − 0.963i)10-s + (0.996 + 0.0836i)11-s + (0.228 − 0.973i)12-s + (0.387 + 0.921i)13-s + (−0.348 + 0.937i)14-s + (−0.268 + 0.963i)15-s + (0.913 − 0.406i)16-s + (0.832 + 0.553i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.912 - 0.409i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.912 - 0.409i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(151\)
\( \varepsilon \)  =  $0.912 - 0.409i$
motivic weight  =  \(0\)
character  :  $\chi_{151} (43, \cdot )$
Sato-Tate  :  $\mu(75)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 151,\ (0:\ ),\ 0.912 - 0.409i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.9222733149 - 0.1974461875i$
$L(\frac12,\chi)$  $\approx$  $0.9222733149 - 0.1974461875i$
$L(\chi,1)$  $\approx$  0.9044220493 - 0.1891369668i
$L(1,\chi)$  $\approx$  0.9044220493 - 0.1891369668i

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

−28.159676988216192423148077264453, −27.07551720775960504666017511714, −25.608984471044964846657891769943, −25.15650500910265335395815043728, −24.649367911183914271531473677468, −23.058603224442331885485157545903, −22.68000070742071900973927247382, −21.68441060995191229597754275640, −19.90731772561957655440880666341, −18.56895038362677964795583960886, −18.29033843784953448887528610404, −16.95189143183638609991962738999, −16.503374989652908895622583086767, −14.92539431163866191433207334552, −13.9101034680947800491266867376, −13.080198613800292728955770628292, −12.13927246534267783469785267348, −10.33279748171799833654822150523, −9.31778567228206066606146963344, −8.1190960835222206282620590233, −6.75768453964001681635537656842, −6.123685084247653279918839850668, −5.273564533859426907483650470017, −3.12250502486134481240698894148, −1.18707689008955803108642144917, 1.30859266317901698156440068646, 3.13426081291628277380010249427, 4.146557029951416452671009560, 5.417434234376918358898438211681, 6.6588462311334575814515007188, 9.020371895503115326780627725831, 9.44924782866893359046312951716, 10.40458014988690590978880764288, 11.407326151546535619891399278532, 12.5669450212492913211563432606, 13.68334698235049371923595665519, 14.59608872735134676341414681047, 16.36210556762770354080085664127, 17.019440285326157745041976320847, 17.92306959005312215965359200632, 19.33011802395056201399054658293, 20.17168297389752181228944856792, 21.3464077080735004646724853231, 21.77117021609406535641196169319, 22.73244370540211030721611727507, 23.66567907764344721206494564017, 25.567575091530979344970375492916, 26.15862882744817518588571359817, 27.24703659035659261681164482571, 28.19504494841372558930406447651

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