↑

Properties

Label 1-151-151.144-r0-0-0
Degree $1$
Conductor $151$
Sign $0.912 + 0.409i$
Analytic cond. $0.701241$
Root an. cond. $0.701241$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(151\)
Sign: $0.912 + 0.409i$
Analytic conductor: \(0.701241\)
Root analytic conductor: \(0.701241\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{151} (144, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 151,\ (0:\ ),\ 0.912 + 0.409i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9222733149 + 0.1974461875i\)
\(L(\frac12)\) \(\approx\) \(0.9222733149 + 0.1974461875i\)
\(L(1)\) \(\approx\) \(0.9044220493 + 0.1891369668i\)
\(L(1)\) \(\approx\) \(0.9044220493 + 0.1891369668i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad151 \( 1 \)
good2 \( 1 + (-0.104 + 0.994i)T \)
3 \( 1 + (-0.425 - 0.904i)T \)
5 \( 1 + (0.985 + 0.166i)T \)
7 \( 1 + (-0.895 + 0.444i)T \)
11 \( 1 + (0.996 - 0.0836i)T \)
13 \( 1 + (0.387 - 0.921i)T \)
17 \( 1 + (0.832 - 0.553i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (0.669 - 0.743i)T \)
29 \( 1 + (0.876 - 0.481i)T \)
31 \( 1 + (-0.999 - 0.0418i)T \)
37 \( 1 + (0.604 + 0.796i)T \)
41 \( 1 + (-0.187 + 0.982i)T \)
43 \( 1 + (-0.895 - 0.444i)T \)
47 \( 1 + (-0.756 - 0.653i)T \)
53 \( 1 + (0.728 + 0.684i)T \)
59 \( 1 + (-0.809 + 0.587i)T \)
61 \( 1 + (-0.570 - 0.821i)T \)
67 \( 1 + (-0.637 - 0.770i)T \)
71 \( 1 + (0.832 + 0.553i)T \)
73 \( 1 + (0.0627 - 0.998i)T \)
79 \( 1 + (0.535 + 0.844i)T \)
83 \( 1 + (-0.929 + 0.368i)T \)
89 \( 1 + (0.146 - 0.989i)T \)
97 \( 1 + (-0.348 + 0.937i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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