Properties

Degree 1
Conductor $ 7 \cdot 11 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  − 2-s − 3-s + 4-s − 5-s + 6-s − 8-s + 9-s + 10-s − 12-s + 13-s + 15-s + 16-s + 17-s − 18-s + 19-s − 20-s + 23-s + 24-s + 25-s − 26-s − 27-s − 29-s − 30-s − 31-s − 32-s − 34-s + 36-s + ⋯
L(s,χ)  = 1  − 2-s − 3-s + 4-s − 5-s + 6-s − 8-s + 9-s + 10-s − 12-s + 13-s + 15-s + 16-s + 17-s − 18-s + 19-s − 20-s + 23-s + 24-s + 25-s − 26-s − 27-s − 29-s − 30-s − 31-s − 32-s − 34-s + 36-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(77\)    =    \(7 \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{77} (76, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(1,\ 77,\ (0:\ ),\ 1)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.4182400222$
$L(\frac12,\chi)$  $\approx$  $0.4182400222$
$L(\chi,1)$  $\approx$  0.4979265317
$L(1,\chi)$  $\approx$  0.4979265317

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

−30.94277776709996655356941385830, −29.95368946866670198730352592417, −28.8276077041774636962363391404, −27.925588437926485895454634834449, −27.28930991398002848192886678524, −26.19771598585890605169585061368, −24.77709810862174352592399878500, −23.69732075073176553842644583995, −22.84615655212071405648302769446, −21.288914672040234576373997068043, −20.152781805949400179553861939395, −18.81373089363854149644084108168, −18.20251946705130657171112613701, −16.73559968481142131010888879651, −16.12326360357351206788546607880, −15.0079111508592099644854016147, −12.73546900626758556607578222035, −11.54828713233477550133866120260, −10.927311664284949109442757075780, −9.49909710522210984532320433497, −7.95830051980204677705751716148, −6.94388387084431951400558554216, −5.50311450515315147546139640413, −3.55056670578384210285742013662, −1.07518664972339127935611675580, 1.07518664972339127935611675580, 3.55056670578384210285742013662, 5.50311450515315147546139640413, 6.94388387084431951400558554216, 7.95830051980204677705751716148, 9.49909710522210984532320433497, 10.927311664284949109442757075780, 11.54828713233477550133866120260, 12.73546900626758556607578222035, 15.0079111508592099644854016147, 16.12326360357351206788546607880, 16.73559968481142131010888879651, 18.20251946705130657171112613701, 18.81373089363854149644084108168, 20.152781805949400179553861939395, 21.288914672040234576373997068043, 22.84615655212071405648302769446, 23.69732075073176553842644583995, 24.77709810862174352592399878500, 26.19771598585890605169585061368, 27.28930991398002848192886678524, 27.925588437926485895454634834449, 28.8276077041774636962363391404, 29.95368946866670198730352592417, 30.94277776709996655356941385830

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