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Properties

Label 1-77-77.76-r0-0-0
Degree $1$
Conductor $77$
Sign $1$
Analytic cond. $0.357586$
Root an. cond. $0.357586$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $1$
Analytic conductor: \(0.357586\)
Root analytic conductor: \(0.357586\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{77} (76, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 77,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4182400222\)
\(L(\frac12)\) \(\approx\) \(0.4182400222\)
\(L(1)\) \(\approx\) \(0.4979265317\)
\(L(1)\) \(\approx\) \(0.4979265317\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 - T \)
13 \( 1 + T \)
17 \( 1 + T \)
19 \( 1 + T \)
23 \( 1 + T \)
29 \( 1 - T \)
31 \( 1 - T \)
37 \( 1 + T \)
41 \( 1 + T \)
43 \( 1 - T \)
47 \( 1 - T \)
53 \( 1 + T \)
59 \( 1 - T \)
61 \( 1 + T \)
67 \( 1 + T \)
71 \( 1 + T \)
73 \( 1 + T \)
79 \( 1 - T \)
83 \( 1 + T \)
89 \( 1 - T \)
97 \( 1 - T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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