Properties

Degree 1
Conductor 107
Sign $0.457 + 0.888i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.998 + 0.0592i)2-s + (−0.533 + 0.845i)3-s + (0.992 − 0.118i)4-s + (0.937 + 0.348i)5-s + (0.482 − 0.875i)6-s + (0.829 + 0.558i)7-s + (−0.984 + 0.176i)8-s + (−0.430 − 0.902i)9-s + (−0.956 − 0.292i)10-s + (0.263 − 0.964i)11-s + (−0.430 + 0.902i)12-s + (0.674 − 0.737i)13-s + (−0.861 − 0.508i)14-s + (−0.794 + 0.606i)15-s + (0.972 − 0.234i)16-s + (0.147 + 0.989i)17-s + ⋯
L(s,χ)  = 1  + (−0.998 + 0.0592i)2-s + (−0.533 + 0.845i)3-s + (0.992 − 0.118i)4-s + (0.937 + 0.348i)5-s + (0.482 − 0.875i)6-s + (0.829 + 0.558i)7-s + (−0.984 + 0.176i)8-s + (−0.430 − 0.902i)9-s + (−0.956 − 0.292i)10-s + (0.263 − 0.964i)11-s + (−0.430 + 0.902i)12-s + (0.674 − 0.737i)13-s + (−0.861 − 0.508i)14-s + (−0.794 + 0.606i)15-s + (0.972 − 0.234i)16-s + (0.147 + 0.989i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.457 + 0.888i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.457 + 0.888i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(107\)
\( \varepsilon \)  =  $0.457 + 0.888i$
motivic weight  =  \(0\)
character  :  $\chi_{107} (53, \cdot )$
Sato-Tate  :  $\mu(53)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 107,\ (0:\ ),\ 0.457 + 0.888i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.6164257649 + 0.3758581321i$
$L(\frac12,\chi)$  $\approx$  $0.6164257649 + 0.3758581321i$
$L(\chi,1)$  $\approx$  0.7033422302 + 0.2557641638i
$L(1,\chi)$  $\approx$  0.7033422302 + 0.2557641638i

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

−29.39597250315226240727566599515, −28.28705849423604311535961139490, −27.820737889645813668707105125916, −26.22542561112237605042827768931, −25.39272209557716054453139708974, −24.41647090618174304022548993917, −23.69048192862706363749151827697, −22.12655613377944987740729257173, −20.768369811911682116262172673613, −20.08157676339654060877333467533, −18.576073759946348482055741780, −17.86290037034408329155605272558, −17.17201294806989353451691333934, −16.2441879527062908942944284517, −14.42538771346298667389214917116, −13.28093084382415639684592154910, −11.953325739561595451224911658391, −11.04015692428073630702969270075, −9.79984243832565248212771684198, −8.56571090946017593110776430032, −7.27924198649171774896690267286, −6.406039722712832398389122960900, −4.89300191582775697413834338101, −2.1938988358830094445732178221, −1.258576809761804593398115503558, 1.6401249323484179750186185279, 3.40829058900764277136358212072, 5.66222285099428261214275392422, 6.0917430276722056591925858485, 8.135710566746853713558317504077, 9.111030046383538797088934948961, 10.355707911488651584354567571145, 10.954249623964858991444318120, 12.16574367209966244225955946066, 14.185773906420664345994056610289, 15.204692943108227721507906083242, 16.27002448890765643936811146283, 17.39193991926430377168716730015, 17.945999911749067688048411218673, 19.10211406597098908650367685804, 20.77360220046296434335217862984, 21.25839650668272547139037203312, 22.2635712889275610501464554324, 23.83006282921252889199316030328, 24.965218121711517004940529229581, 25.88318665732653223612145378385, 26.90317072372216947243686721816, 27.73004109453442407904122097647, 28.51483857640230811980182007293, 29.526376413096748654615310502156

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