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Properties

Label 1-244-244.243-r1-0-0
Degree $1$
Conductor $244$
Sign $1$
Analytic cond. $26.2214$
Root an. cond. $26.2214$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 244 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 244 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(244\)    =    \(2^{2} \cdot 61\)
Sign: $1$
Analytic conductor: \(26.2214\)
Root analytic conductor: \(26.2214\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{244} (243, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 244,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.095803692\)
\(L(\frac12)\) \(\approx\) \(2.095803692\)
\(L(1)\) \(\approx\) \(1.206719164\)
\(L(1)\) \(\approx\) \(1.206719164\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
61 \( 1 \)
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 \( 1 + T \)
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

−25.8458231493869993033920071492, −24.71461540810001718574713762363, −24.22624989102474586516302605546, −23.007922214094401359841737281806, −22.25906233964061351420242230658, −21.25617877546655158897978647797, −20.78379740378462617564909553920, −19.18532654790545831133219967708, −18.139961792146404794292759293043, −17.40456828990801349795438227839, −16.92502276308245764161865411738, −15.60865594538199558107588897197, −14.525408630233256658119226615707, −13.4798340853100170187594285006, −12.55681844899534317299928374977, −11.198071276220859387886651019471, −10.87914159548201251408798094371, −9.48825467883900708079804416763, −8.51187813590284935165535011638, −6.86790419879404366728884558390, −6.15253662598808775176088968586, −5.07910104725054612833276243337, −4.07838889627063651840952005547, −2.00931901390469847833147039966, −1.05571736595104576017152691872, 1.05571736595104576017152691872, 2.00931901390469847833147039966, 4.07838889627063651840952005547, 5.07910104725054612833276243337, 6.15253662598808775176088968586, 6.86790419879404366728884558390, 8.51187813590284935165535011638, 9.48825467883900708079804416763, 10.87914159548201251408798094371, 11.198071276220859387886651019471, 12.55681844899534317299928374977, 13.4798340853100170187594285006, 14.525408630233256658119226615707, 15.60865594538199558107588897197, 16.92502276308245764161865411738, 17.40456828990801349795438227839, 18.139961792146404794292759293043, 19.18532654790545831133219967708, 20.78379740378462617564909553920, 21.25617877546655158897978647797, 22.25906233964061351420242230658, 23.007922214094401359841737281806, 24.22624989102474586516302605546, 24.71461540810001718574713762363, 25.8458231493869993033920071492

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