Properties

Label 1-151-151.11-r0-0-0
Degree $1$
Conductor $151$
Sign $-0.350 - 0.936i$
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  + (0.669 − 0.743i)2-s + (0.0627 − 0.998i)3-s + (−0.104 − 0.994i)4-s + (0.387 + 0.921i)5-s + (−0.699 − 0.714i)6-s + (0.996 − 0.0836i)7-s + (−0.809 − 0.587i)8-s + (−0.992 − 0.125i)9-s + (0.944 + 0.328i)10-s + (0.832 − 0.553i)11-s + (−0.999 + 0.0418i)12-s + (−0.348 − 0.937i)13-s + (0.604 − 0.796i)14-s + (0.944 − 0.328i)15-s + (−0.978 + 0.207i)16-s + (−0.570 + 0.821i)17-s + ⋯
L(s)  = 1  + (0.669 − 0.743i)2-s + (0.0627 − 0.998i)3-s + (−0.104 − 0.994i)4-s + (0.387 + 0.921i)5-s + (−0.699 − 0.714i)6-s + (0.996 − 0.0836i)7-s + (−0.809 − 0.587i)8-s + (−0.992 − 0.125i)9-s + (0.944 + 0.328i)10-s + (0.832 − 0.553i)11-s + (−0.999 + 0.0418i)12-s + (−0.348 − 0.937i)13-s + (0.604 − 0.796i)14-s + (0.944 − 0.328i)15-s + (−0.978 + 0.207i)16-s + (−0.570 + 0.821i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(151\)
Sign: $-0.350 - 0.936i$
Analytic conductor: \(0.701241\)
Root analytic conductor: \(0.701241\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{151} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 151,\ (0:\ ),\ -0.350 - 0.936i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9591988354 - 1.383485654i\)
\(L(\frac12)\) \(\approx\) \(0.9591988354 - 1.383485654i\)
\(L(1)\) \(\approx\) \(1.202486104 - 0.9729723884i\)
\(L(1)\) \(\approx\) \(1.202486104 - 0.9729723884i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad151 \( 1 \)
good2 \( 1 + (0.669 - 0.743i)T \)
3 \( 1 + (0.0627 - 0.998i)T \)
5 \( 1 + (0.387 + 0.921i)T \)
7 \( 1 + (0.996 - 0.0836i)T \)
11 \( 1 + (0.832 - 0.553i)T \)
13 \( 1 + (-0.348 - 0.937i)T \)
17 \( 1 + (-0.570 + 0.821i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (0.913 + 0.406i)T \)
29 \( 1 + (-0.929 + 0.368i)T \)
31 \( 1 + (-0.957 - 0.289i)T \)
37 \( 1 + (0.985 + 0.166i)T \)
41 \( 1 + (0.968 - 0.248i)T \)
43 \( 1 + (0.996 + 0.0836i)T \)
47 \( 1 + (-0.268 + 0.963i)T \)
53 \( 1 + (0.535 - 0.844i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 + (-0.895 - 0.444i)T \)
67 \( 1 + (-0.992 + 0.125i)T \)
71 \( 1 + (-0.570 - 0.821i)T \)
73 \( 1 + (-0.425 + 0.904i)T \)
79 \( 1 + (0.728 + 0.684i)T \)
83 \( 1 + (0.876 + 0.481i)T \)
89 \( 1 + (-0.855 + 0.518i)T \)
97 \( 1 + (0.604 + 0.796i)T \)
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

−28.080307791951937215582281187958, −27.31155889789618409261855434340, −26.355427538602326105833056622463, −25.22406843884982556558443895360, −24.54442284118215786741909402550, −23.54517098790999711954042320078, −22.36088673702397194061965771387, −21.47277161384581029162218220192, −20.859920362245316917504150792621, −19.920809642211444530209959098815, −17.84765841579490890170719856026, −16.98555843229332489650185762213, −16.40939206908503711067065479265, −15.09040743660898422676659708190, −14.48739255906880121939248265738, −13.40196713368923840975335527971, −12.05863126012647910525114760825, −11.14839292472806268218637415976, −9.1819562625038843350088387988, −8.89241570535633390382895213207, −7.31386594924064814835204793902, −5.818734856972443219137628947099, −4.60360016766279797104936105368, −4.34205130262080966573660005527, −2.28831594144626107283289563596, 1.42858331402699211497507398540, 2.46742545188370684766465068608, 3.76807304417996391179730374295, 5.536656525109091126462052435864, 6.39517886502305057167380772817, 7.70110149348817380726750855267, 9.137127621241662530424959393, 10.82652876961973252233879351349, 11.22500323309256682752476953749, 12.56721455628567648677129713102, 13.43620505726719279294286846827, 14.66184429373034741764899494161, 14.7748308529216571100505962847, 17.199623852463465945415912303952, 18.00557046692380115680131082844, 18.97843653021864962146549241203, 19.73108123194625486935275788174, 20.896837125548646794265551086522, 21.963185727440706920591839139486, 22.75286489347103215604208128236, 23.80517732555394721765498839039, 24.59148355920164656579660161924, 25.51822216933146787090551556052, 27.04206890877585074689111711541, 27.86658750233366537362156687980

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