Properties

Degree 1
Conductor $ 7^{2} $
Sign $0.942 + 0.335i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.826 + 0.563i)2-s + (−0.733 − 0.680i)3-s + (0.365 + 0.930i)4-s + (0.955 − 0.294i)5-s + (−0.222 − 0.974i)6-s + (−0.222 + 0.974i)8-s + (0.0747 + 0.997i)9-s + (0.955 + 0.294i)10-s + (0.0747 − 0.997i)11-s + (0.365 − 0.930i)12-s + (−0.900 + 0.433i)13-s + (−0.900 − 0.433i)15-s + (−0.733 + 0.680i)16-s + (−0.988 + 0.149i)17-s + (−0.5 + 0.866i)18-s + (−0.5 − 0.866i)19-s + ⋯
L(s,χ)  = 1  + (0.826 + 0.563i)2-s + (−0.733 − 0.680i)3-s + (0.365 + 0.930i)4-s + (0.955 − 0.294i)5-s + (−0.222 − 0.974i)6-s + (−0.222 + 0.974i)8-s + (0.0747 + 0.997i)9-s + (0.955 + 0.294i)10-s + (0.0747 − 0.997i)11-s + (0.365 − 0.930i)12-s + (−0.900 + 0.433i)13-s + (−0.900 − 0.433i)15-s + (−0.733 + 0.680i)16-s + (−0.988 + 0.149i)17-s + (−0.5 + 0.866i)18-s + (−0.5 − 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.942 + 0.335i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.942 + 0.335i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(49\)    =    \(7^{2}\)
\( \varepsilon \)  =  $0.942 + 0.335i$
motivic weight  =  \(0\)
character  :  $\chi_{49} (2, \cdot )$
Sato-Tate  :  $\mu(21)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 49,\ (0:\ ),\ 0.942 + 0.335i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.122815745 + 0.1938615830i$
$L(\frac12,\chi)$  $\approx$  $1.122815745 + 0.1938615830i$
$L(\chi,1)$  $\approx$  1.280466911 + 0.1806316226i
$L(1,\chi)$  $\approx$  1.280466911 + 0.1806316226i

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

−33.52325351169627776602221447245, −32.74631799844587404996258560454, −31.61467828956052643899433273280, −30.12109595893963834065689889641, −29.219700139528940594552965133898, −28.368839807944114467393672477788, −27.15709383241467038400905555058, −25.54698009659237326018041340098, −24.20598491182213247613786569680, −22.74698673172678379862502231044, −22.176714330660254953078481240760, −21.109587981789763952545930366350, −20.065815936928411652954614701835, −18.22384584387758560590029146188, −17.10766479957415847740792639573, −15.43997528020018259022189405218, −14.480764046614879481990767539, −12.947518065591294161054323178972, −11.76618446952053708685270123667, −10.318736691759783264319546253761, −9.69962263339120392198709477488, −6.65890100171355105377608803012, −5.47640029502383725092379230024, −4.2263532877540372808585200766, −2.24339800929317109496992518107, 2.29455115051436058876020397693, 4.75403887340930169710031481621, 5.956319884684112567375983290985, 6.95438298346737261907270838384, 8.693319548284248728291819716807, 10.87798358591115479113488115584, 12.2605184342386236118323180501, 13.31569894688231298831225380389, 14.23381942366225381251726984000, 16.11921480034142001995943515043, 17.093602146792523730652118848442, 18.007496912436223229873483196186, 19.80789257991871975676818409348, 21.72961554918514272153972861333, 21.988045959991187279434883995742, 23.71580821367981595569932653160, 24.36005746307786329279093367574, 25.337739661909230008336026520459, 26.72273894166387383440253122905, 28.61335759319650900622284794465, 29.474373037088049719356341633936, 30.33800838877746614975288374930, 31.76872881951831081908138836272, 32.84547753665400128967984795879, 33.91181740593369290783369780544

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