Properties

Degree 1
Conductor 349
Sign $0.751 + 0.659i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.935 + 0.353i)2-s + (−0.999 + 0.0361i)3-s + (0.750 + 0.661i)4-s + (0.958 + 0.284i)5-s + (−0.947 − 0.319i)6-s + (0.590 − 0.806i)7-s + (0.468 + 0.883i)8-s + (0.997 − 0.0721i)9-s + (0.796 + 0.605i)10-s + (0.267 + 0.963i)11-s + (−0.773 − 0.633i)12-s + (−0.891 − 0.452i)13-s + (0.837 − 0.546i)14-s + (−0.968 − 0.250i)15-s + (0.126 + 0.992i)16-s + (0.468 − 0.883i)17-s + ⋯
L(s,χ)  = 1  + (0.935 + 0.353i)2-s + (−0.999 + 0.0361i)3-s + (0.750 + 0.661i)4-s + (0.958 + 0.284i)5-s + (−0.947 − 0.319i)6-s + (0.590 − 0.806i)7-s + (0.468 + 0.883i)8-s + (0.997 − 0.0721i)9-s + (0.796 + 0.605i)10-s + (0.267 + 0.963i)11-s + (−0.773 − 0.633i)12-s + (−0.891 − 0.452i)13-s + (0.837 − 0.546i)14-s + (−0.968 − 0.250i)15-s + (0.126 + 0.992i)16-s + (0.468 − 0.883i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.751 + 0.659i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.751 + 0.659i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(349\)
\( \varepsilon \)  =  $0.751 + 0.659i$
motivic weight  =  \(0\)
character  :  $\chi_{349} (180, \cdot )$
Sato-Tate  :  $\mu(87)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 349,\ (0:\ ),\ 0.751 + 0.659i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.975927380 + 0.7444103245i$
$L(\frac12,\chi)$  $\approx$  $1.975927380 + 0.7444103245i$
$L(\chi,1)$  $\approx$  1.618291116 + 0.4208572669i
$L(1,\chi)$  $\approx$  1.618291116 + 0.4208572669i

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

−24.44151948725457647028166019274, −24.00443107984745150621564144635, −22.77672858281375987086072075325, −21.92253737296938096622766187249, −21.476457659729703782350531567928, −20.880054500855743348595226020673, −19.354970565752533142883390078524, −18.60431971610491137622454000531, −17.49652023849197748384615266790, −16.67760623137797884820981259744, −15.80132156200586970785204806724, −14.57615574815463006779004650792, −13.90918115886586262882196953138, −12.71355750859246165347010952218, −12.094893911775709347209379409200, −11.29291777613898358922384899909, −10.26898167823672558147061821367, −9.42017489608840969233934235849, −7.81145400301388348472232491936, −6.28080024898363471077646765024, −5.78593137626546064141675507234, −5.06457718246371742996108605347, −3.88443889395475495657563505612, −2.23263337498643792284416305571, −1.382189189822857105722021249453, 1.49545165303549240898243296196, 2.793665764909179286846576232796, 4.492511525218399934974927117371, 4.956273425151800467865422564, 6.03606711208107763562039373333, 7.02993559023578708668459965641, 7.59635663429424726465713522308, 9.62771394537232798006501045370, 10.437790749212156451877748185253, 11.41593619109575608282951602614, 12.29704459240804491250420431211, 13.20890551243477406764417715964, 14.12586035392415561435705421708, 14.90524190592538069455353806015, 16.00238191014704300175514102911, 17.13425825309913509085988767090, 17.39099831762406919765291201643, 18.30730841993409034765757098213, 20.07968718546694377531055008448, 20.745555092371878693102100613, 21.73517688608064428372397447894, 22.45207184088623430649622878503, 22.95348972770435602659403452011, 24.10347646082447263565418670283, 24.57522739318766603239457195140

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