Properties

Degree 1
Conductor 151
Sign $-0.350 + 0.936i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(151\)
\( \varepsilon \)  =  $-0.350 + 0.936i$
motivic weight  =  \(0\)
character  :  $\chi_{151} (55, \cdot )$
Sato-Tate  :  $\mu(75)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 151,\ (0:\ ),\ -0.350 + 0.936i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.9591988354 + 1.383485654i$
$L(\frac12,\chi)$  $\approx$  $0.9591988354 + 1.383485654i$
$L(\chi,1)$  $\approx$  1.202486104 + 0.9729723884i
$L(1,\chi)$  $\approx$  1.202486104 + 0.9729723884i

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

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

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