Properties

Label 1-56-56.13-r1-0-0
Degree $1$
Conductor $56$
Sign $1$
Analytic cond. $6.01803$
Root an. cond. $6.01803$
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 + 9-s − 11-s + 13-s + 15-s − 17-s + 19-s + 23-s + 25-s + 27-s − 29-s − 31-s − 33-s − 37-s + 39-s − 41-s − 43-s + 45-s − 47-s − 51-s − 53-s − 55-s + 57-s + 59-s + 61-s + 65-s + ⋯
L(s)  = 1  + 3-s + 5-s + 9-s − 11-s + 13-s + 15-s − 17-s + 19-s + 23-s + 25-s + 27-s − 29-s − 31-s − 33-s − 37-s + 39-s − 41-s − 43-s + 45-s − 47-s − 51-s − 53-s − 55-s + 57-s + 59-s + 61-s + 65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(56\)    =    \(2^{3} \cdot 7\)
Sign: $1$
Analytic conductor: \(6.01803\)
Root analytic conductor: \(6.01803\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{56} (13, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 56,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.447541817\)
\(L(\frac12)\) \(\approx\) \(2.447541817\)
\(L(1)\) \(\approx\) \(1.679251908\)
\(L(1)\) \(\approx\) \(1.679251908\)

Euler product

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

−32.9010348270297627088892070871, −31.51944231943997184472652470117, −30.690000746945030200012261744824, −29.43000418345294518925026094548, −28.41414520960288211549091183390, −26.75231591592850034861396584812, −25.92960509169980967974778911023, −25.01828837110298869760597156425, −23.9105265517369983533216599615, −22.22697215961054858321994589585, −21.008822935727199961141204299343, −20.35867295442329041893126835339, −18.73788426990777456133268263405, −17.89852150528378451400371119022, −16.17944515506529237932446357855, −14.98534622000614502074559353283, −13.60126566450937089198195650280, −13.06130986325124306941443320343, −10.854841016737571399498258970083, −9.56977863791393047057245889649, −8.48938031650563658288632406702, −6.93325712626592541726906678836, −5.20981502796481458623915670533, −3.252991709700736450463186226578, −1.789478909894140664437254075447, 1.789478909894140664437254075447, 3.252991709700736450463186226578, 5.20981502796481458623915670533, 6.93325712626592541726906678836, 8.48938031650563658288632406702, 9.56977863791393047057245889649, 10.854841016737571399498258970083, 13.06130986325124306941443320343, 13.60126566450937089198195650280, 14.98534622000614502074559353283, 16.17944515506529237932446357855, 17.89852150528378451400371119022, 18.73788426990777456133268263405, 20.35867295442329041893126835339, 21.008822935727199961141204299343, 22.22697215961054858321994589585, 23.9105265517369983533216599615, 25.01828837110298869760597156425, 25.92960509169980967974778911023, 26.75231591592850034861396584812, 28.41414520960288211549091183390, 29.43000418345294518925026094548, 30.690000746945030200012261744824, 31.51944231943997184472652470117, 32.9010348270297627088892070871

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