Properties

Degree $1$
Conductor $149$
Sign $-0.835 - 0.549i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.127 + 0.991i)2-s + (−0.996 − 0.0848i)3-s + (−0.967 − 0.251i)4-s + (0.778 + 0.628i)5-s + (0.210 − 0.977i)6-s + (−0.721 + 0.691i)7-s + (0.372 − 0.927i)8-s + (0.985 + 0.169i)9-s + (−0.721 + 0.691i)10-s + (−0.967 + 0.251i)11-s + (0.942 + 0.333i)12-s + (−0.450 + 0.892i)13-s + (−0.594 − 0.803i)14-s + (−0.721 − 0.691i)15-s + (0.873 + 0.487i)16-s + (−0.996 − 0.0848i)17-s + ⋯
L(s,χ)  = 1  + (−0.127 + 0.991i)2-s + (−0.996 − 0.0848i)3-s + (−0.967 − 0.251i)4-s + (0.778 + 0.628i)5-s + (0.210 − 0.977i)6-s + (−0.721 + 0.691i)7-s + (0.372 − 0.927i)8-s + (0.985 + 0.169i)9-s + (−0.721 + 0.691i)10-s + (−0.967 + 0.251i)11-s + (0.942 + 0.333i)12-s + (−0.450 + 0.892i)13-s + (−0.594 − 0.803i)14-s + (−0.721 − 0.691i)15-s + (0.873 + 0.487i)16-s + (−0.996 − 0.0848i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.835 - 0.549i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.835 - 0.549i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(149\)
Sign: $-0.835 - 0.549i$
Motivic weight: \(0\)
Character: $\chi_{149} (95, \cdot )$
Sato-Tate group: $\mu(37)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 149,\ (0:\ ),\ -0.835 - 0.549i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(-0.09084161750 + 0.3034779715i\)
\(L(\frac12,\chi)\) \(\approx\) \(-0.09084161750 + 0.3034779715i\)
\(L(\chi,1)\) \(\approx\) \(0.3887537841 + 0.3635483941i\)
\(L(1,\chi)\) \(\approx\) \(0.3887537841 + 0.3635483941i\)

Euler product

   \(L(\chi,s) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)
   \(L(s,\chi) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−27.84256428919243077067609930108, −26.846620819092559072621722918801, −25.855574262328108964819851402807, −24.37094471080456091308892288494, −23.29604795652868355430390008639, −22.48201394910455329061017532466, −21.57749940909755196907236529953, −20.703630057582350649792131181534, −19.73496816551328799431878164375, −18.43760017185870732925328567685, −17.54738996143345067714721371341, −16.86484422164631732877066653331, −15.72528060033662451965922117268, −13.753299411939622982315443019419, −12.95024900958774171228015448911, −12.31209566323071174217199973042, −10.732547274557831966255007248179, −10.26320898023285024295133419569, −9.21959267959405868008214840633, −7.63411820584168089102183510011, −5.90374435557641549697845161744, −5.012503371152348342744433082766, −3.67882598868501763171926874533, −1.92868668489003036263806971796, −0.30706036233468212580079303282, 2.31565804695403734279895693820, 4.51277394342273278288732290338, 5.63489687773273199250875609846, 6.48611452052076835268370797605, 7.257157307399663257381283495044, 9.07544044993744768380013995032, 9.94680127530476614452034445205, 11.06636155038894511802735202926, 12.69982035251017056991749253519, 13.39213836714388921945688390733, 14.81820945127152654367278356376, 15.76572872614433430955470449771, 16.65268954330428932916778069388, 17.73773572400362470192052694123, 18.34658771402232744159433571775, 19.198476330417945160588777529552, 21.33372423609547612748960615737, 22.2236025862488344832622397383, 22.663662338796076792762529710148, 23.986488207002292475079439504262, 24.59933158836541069193610508668, 26.141733794756930653051840402730, 26.182264919735567706800137048373, 27.8012782522696301018956845175, 28.678779413056156668848584669900

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