Properties

Label 1-772-772.771-r1-0-0
Degree $1$
Conductor $772$
Sign $1$
Analytic cond. $82.9629$
Root an. cond. $82.9629$
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  − 3-s − 5-s − 7-s + 9-s + 11-s − 13-s + 15-s − 17-s + 19-s + 21-s − 23-s + 25-s − 27-s − 29-s − 31-s − 33-s + 35-s − 37-s + 39-s − 41-s − 43-s − 45-s + 47-s + 49-s + 51-s − 53-s − 55-s + ⋯
L(s)  = 1  − 3-s − 5-s − 7-s + 9-s + 11-s − 13-s + 15-s − 17-s + 19-s + 21-s − 23-s + 25-s − 27-s − 29-s − 31-s − 33-s + 35-s − 37-s + 39-s − 41-s − 43-s − 45-s + 47-s + 49-s + 51-s − 53-s − 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 772 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 772 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(772\)    =    \(2^{2} \cdot 193\)
Sign: $1$
Analytic conductor: \(82.9629\)
Root analytic conductor: \(82.9629\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{772} (771, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 772,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2541609444\)
\(L(\frac12)\) \(\approx\) \(0.2541609444\)
\(L(1)\) \(\approx\) \(0.4522735748\)
\(L(1)\) \(\approx\) \(0.4522735748\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
193 \( 1 \)
good3 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 + T \)
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

−22.340826070947383987420166779162, −21.860007109800059699709171466112, −20.21369521994346419742775222446, −19.83542002071662439956077419771, −18.91938761517351916883407284613, −18.19407619844604721614661815984, −17.080429659784414964867142467094, −16.58276830894488479990651966518, −15.74250710540369677365882570253, −15.157863512975154064695235110051, −13.94464368278904202622634924428, −12.85103371621734987677023090856, −12.11109946611405080670159135470, −11.64649330652253939655886128126, −10.67925775299916035354455023253, −9.72668335871354168874433546522, −8.970235468572964917849240802144, −7.51078923463509361698624590500, −6.971511436594690974034909898800, −6.11725977919979379827362296539, −5.02240355183018484778416213649, −4.08767881428328631711349264711, −3.312114465973292058215577875912, −1.71600544006715330197888345843, −0.25683062111492527032802653093, 0.25683062111492527032802653093, 1.71600544006715330197888345843, 3.312114465973292058215577875912, 4.08767881428328631711349264711, 5.02240355183018484778416213649, 6.11725977919979379827362296539, 6.971511436594690974034909898800, 7.51078923463509361698624590500, 8.970235468572964917849240802144, 9.72668335871354168874433546522, 10.67925775299916035354455023253, 11.64649330652253939655886128126, 12.11109946611405080670159135470, 12.85103371621734987677023090856, 13.94464368278904202622634924428, 15.157863512975154064695235110051, 15.74250710540369677365882570253, 16.58276830894488479990651966518, 17.080429659784414964867142467094, 18.19407619844604721614661815984, 18.91938761517351916883407284613, 19.83542002071662439956077419771, 20.21369521994346419742775222446, 21.860007109800059699709171466112, 22.340826070947383987420166779162

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