Properties

Degree 1
Conductor $ 7 \cdot 11 $
Sign $0.0395 - 0.999i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.978 + 0.207i)2-s + (−0.104 − 0.994i)3-s + (0.913 − 0.406i)4-s + (0.669 − 0.743i)5-s + (0.309 + 0.951i)6-s + (−0.809 + 0.587i)8-s + (−0.978 + 0.207i)9-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)12-s + (0.309 − 0.951i)13-s + (−0.809 − 0.587i)15-s + (0.669 − 0.743i)16-s + (−0.978 − 0.207i)17-s + (0.913 − 0.406i)18-s + (0.913 + 0.406i)19-s + (0.309 − 0.951i)20-s + ⋯
L(s,χ)  = 1  + (−0.978 + 0.207i)2-s + (−0.104 − 0.994i)3-s + (0.913 − 0.406i)4-s + (0.669 − 0.743i)5-s + (0.309 + 0.951i)6-s + (−0.809 + 0.587i)8-s + (−0.978 + 0.207i)9-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)12-s + (0.309 − 0.951i)13-s + (−0.809 − 0.587i)15-s + (0.669 − 0.743i)16-s + (−0.978 − 0.207i)17-s + (0.913 − 0.406i)18-s + (0.913 + 0.406i)19-s + (0.309 − 0.951i)20-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.0395 - 0.999i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.0395 - 0.999i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(77\)    =    \(7 \cdot 11\)
\( \varepsilon \)  =  $0.0395 - 0.999i$
motivic weight  =  \(0\)
character  :  $\chi_{77} (58, \cdot )$
Sato-Tate  :  $\mu(15)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 77,\ (0:\ ),\ 0.0395 - 0.999i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.4589780525 - 0.4411525607i$
$L(\frac12,\chi)$  $\approx$  $0.4589780525 - 0.4411525607i$
$L(\chi,1)$  $\approx$  0.6476723973 - 0.2985769157i
$L(1,\chi)$  $\approx$  0.6476723973 - 0.2985769157i

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

−31.42419950625557560991030902424, −30.2391149190694466077968694706, −29.01267597237802035788253517126, −28.35801267427729113521190655322, −27.10199320863722619690751530118, −26.25391128660214742663111631951, −25.701118457208868194290113099007, −24.18882937147961157412626360937, −22.39957849752455733144768041223, −21.599965217010592177804059128170, −20.643822960809841932325042757904, −19.44413321996789631775185117384, −18.13955308679429178660593983553, −17.27024389270937391982824799106, −16.10847936282743548538623755600, −15.10383346502125011903092665089, −13.72353871669230264298217521483, −11.636421085167728843044671405497, −10.82159132447325427472777142582, −9.7072349491986737477046949939, −8.90566240860414258671071025466, −7.10214338142543931851792881622, −5.790514889983247207780073431035, −3.72115519101261125281382675087, −2.2232882267185419524740772252, 1.05579301589600281019069120816, 2.47472211027939259189002604237, 5.4481447631619617913231806570, 6.50409605319997825251682704990, 7.91337423597837233491926456012, 8.84584975415468956753847568504, 10.224605654782553723421533785598, 11.66547160483855585945065247982, 12.8556229474564342447183662949, 14.05891444639369134962844692371, 15.72335685613782531630199182112, 16.959757231466721125116036703821, 17.789116798187120151436878795, 18.59661398109408630387540607295, 20.00897997198843423262442436068, 20.61165175582164104614533120, 22.46600825055167831486939390296, 23.96083523785999049294583319083, 24.739385632662065230234680828957, 25.38730309718046578269528442829, 26.62721483633812466652959319854, 28.08012096265825180089638759401, 28.777815273627585115605938447993, 29.62669086965922331256430140957, 30.67659348172208514293503444924

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