↑

Properties

Label 1-151-151.137-r0-0-0
Degree $1$
Conductor $151$
Sign $0.596 - 0.802i$
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  + (0.913 − 0.406i)2-s + (−0.992 + 0.125i)3-s + (0.669 − 0.743i)4-s + (−0.268 + 0.963i)5-s + (−0.855 + 0.518i)6-s + (−0.348 − 0.937i)7-s + (0.309 − 0.951i)8-s + (0.968 − 0.248i)9-s + (0.146 + 0.989i)10-s + (0.604 − 0.796i)11-s + (−0.570 + 0.821i)12-s + (0.944 − 0.328i)13-s + (−0.699 − 0.714i)14-s + (0.146 − 0.989i)15-s + (−0.104 − 0.994i)16-s + (0.985 − 0.166i)17-s + ⋯
L(s)  = 1  + (0.913 − 0.406i)2-s + (−0.992 + 0.125i)3-s + (0.669 − 0.743i)4-s + (−0.268 + 0.963i)5-s + (−0.855 + 0.518i)6-s + (−0.348 − 0.937i)7-s + (0.309 − 0.951i)8-s + (0.968 − 0.248i)9-s + (0.146 + 0.989i)10-s + (0.604 − 0.796i)11-s + (−0.570 + 0.821i)12-s + (0.944 − 0.328i)13-s + (−0.699 − 0.714i)14-s + (0.146 − 0.989i)15-s + (−0.104 − 0.994i)16-s + (0.985 − 0.166i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.596 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.596 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(151\)
Sign: $0.596 - 0.802i$
Analytic conductor: \(0.701241\)
Root analytic conductor: \(0.701241\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{151} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 151,\ (0:\ ),\ 0.596 - 0.802i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.229082179 - 0.6181588807i\)
\(L(\frac12)\) \(\approx\) \(1.229082179 - 0.6181588807i\)
\(L(1)\) \(\approx\) \(1.252376627 - 0.3682248447i\)
\(L(1)\) \(\approx\) \(1.252376627 - 0.3682248447i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad151 \( 1 \)
good2 \( 1 + (0.913 - 0.406i)T \)
3 \( 1 + (-0.992 + 0.125i)T \)
5 \( 1 + (-0.268 + 0.963i)T \)
7 \( 1 + (-0.348 - 0.937i)T \)
11 \( 1 + (0.604 - 0.796i)T \)
13 \( 1 + (0.944 - 0.328i)T \)
17 \( 1 + (0.985 - 0.166i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (-0.978 - 0.207i)T \)
29 \( 1 + (0.728 + 0.684i)T \)
31 \( 1 + (-0.895 - 0.444i)T \)
37 \( 1 + (-0.756 - 0.653i)T \)
41 \( 1 + (0.876 + 0.481i)T \)
43 \( 1 + (-0.348 + 0.937i)T \)
47 \( 1 + (-0.0209 - 0.999i)T \)
53 \( 1 + (-0.425 + 0.904i)T \)
59 \( 1 + (-0.809 + 0.587i)T \)
61 \( 1 + (0.387 + 0.921i)T \)
67 \( 1 + (0.968 + 0.248i)T \)
71 \( 1 + (0.985 + 0.166i)T \)
73 \( 1 + (-0.637 + 0.770i)T \)
79 \( 1 + (0.0627 - 0.998i)T \)
83 \( 1 + (0.535 - 0.844i)T \)
89 \( 1 + (-0.999 - 0.0418i)T \)
97 \( 1 + (-0.699 + 0.714i)T \)
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

−28.28808241791433317161540258581, −27.625902538837535685961735300814, −25.83279855687378116500683099877, −25.0305197192992872183334993872, −24.07304237493724275888662758454, −23.37850570395114689228681493560, −22.43446398777607173016955238807, −21.58358649764324028199258276922, −20.64564451723345234574532094010, −19.36179625848823442113576819941, −17.91621640599990815460980272268, −16.96638367811843572241288180278, −15.9742732572031480609808764857, −15.48882690916183976400925011598, −13.87843076008481016600036023147, −12.62097788165880115882962873104, −12.1855235705347874357745662123, −11.29284552563746381821954036487, −9.52884226017336982253436204999, −8.1653978001349692819524052167, −6.79516747463792539553973468294, −5.78943758641290955713079626443, −4.91350091259897607922998182063, −3.77386708961137074928594000982, −1.72942415173761257197883291564, 1.18544164367369939910070264975, 3.38880828067982445227219857366, 4.003629742968322663818403353732, 5.72817488688563660362129150033, 6.42283318508864861880971073414, 7.53924549288915770270207969391, 9.96365966128544625264765005019, 10.67472329401068260324677492600, 11.45995185678063272883245166527, 12.46867130527797603703258238993, 13.76001322963038070364821374949, 14.533195101149401017791615208760, 15.98408524300028557478706647855, 16.51554577516701589442938848294, 18.14312044839586808411424632108, 19.0000620389985511241548953652, 20.138736716612571262278749700267, 21.33017561086480443540164586706, 22.19653922021059835635305397626, 23.100175977713044467072888461, 23.376556320031696663815081949761, 24.64916506568126663301163120887, 26.00706883689594917746832816201, 27.2163142433974518805986271891, 27.97015638658986081683340952347

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