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Properties

Label 1-69-69.68-r0-0-0
Degree $1$
Conductor $69$
Sign $1$
Analytic cond. $0.320434$
Root an. cond. $0.320434$
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 + 4-s + 5-s − 7-s − 8-s − 10-s + 11-s + 13-s + 14-s + 16-s + 17-s − 19-s + 20-s − 22-s + 25-s − 26-s − 28-s − 29-s + 31-s − 32-s − 34-s − 35-s − 37-s + 38-s − 40-s − 41-s − 43-s + ⋯
L(s)  = 1  − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s + 11-s + 13-s + 14-s + 16-s + 17-s − 19-s + 20-s − 22-s + 25-s − 26-s − 28-s − 29-s + 31-s − 32-s − 34-s − 35-s − 37-s + 38-s − 40-s − 41-s − 43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $1$
Analytic conductor: \(0.320434\)
Root analytic conductor: \(0.320434\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{69} (68, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 69,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6988237658\)
\(L(\frac12)\) \(\approx\) \(0.6988237658\)
\(L(1)\) \(\approx\) \(0.7746280617\)
\(L(1)\) \(\approx\) \(0.7746280617\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
good2 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 - T \)
11 \( 1 + T \)
13 \( 1 + T \)
17 \( 1 + T \)
19 \( 1 - T \)
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

−32.21762384943363625140375589753, −30.18478233071195498047850088711, −29.61957153017785415189247135551, −28.473712795822786583811835280959, −27.674486428848214865235478006793, −26.11883667105154450563610395343, −25.572435838559562173395108703657, −24.6593374485508516502684605759, −23.02537998104311620958290847568, −21.642424019123861664425655738380, −20.61805164647212371890821188456, −19.33905554671463660455604473623, −18.45088290621407564474621445322, −17.13009762136880476636778070362, −16.45807873837363079437020244799, −14.99023247855555059660755042503, −13.483043179299534065516900783939, −12.109009320822192739030152644179, −10.555819577806464845866994174001, −9.600206982670612259211105489811, −8.6100294925213863239155525085, −6.75267169956386058832091004744, −5.95435910875393137127761845837, −3.316091983481942862003250631661, −1.57186578602810803483908737273, 1.57186578602810803483908737273, 3.316091983481942862003250631661, 5.95435910875393137127761845837, 6.75267169956386058832091004744, 8.6100294925213863239155525085, 9.600206982670612259211105489811, 10.555819577806464845866994174001, 12.109009320822192739030152644179, 13.483043179299534065516900783939, 14.99023247855555059660755042503, 16.45807873837363079437020244799, 17.13009762136880476636778070362, 18.45088290621407564474621445322, 19.33905554671463660455604473623, 20.61805164647212371890821188456, 21.642424019123861664425655738380, 23.02537998104311620958290847568, 24.6593374485508516502684605759, 25.572435838559562173395108703657, 26.11883667105154450563610395343, 27.674486428848214865235478006793, 28.473712795822786583811835280959, 29.61957153017785415189247135551, 30.18478233071195498047850088711, 32.21762384943363625140375589753

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