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Properties

Label 1-4956-4956.4955-r0-0-0
Degree $1$
Conductor $4956$
Sign $1$
Analytic cond. $23.0155$
Root an. cond. $23.0155$
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  + 5-s − 11-s + 13-s + 17-s + 19-s − 23-s + 25-s − 29-s − 31-s − 37-s + 41-s + 43-s + 47-s − 53-s − 55-s + 61-s + 65-s + 67-s + 71-s + 73-s − 79-s + 83-s + 85-s − 89-s + 95-s + 97-s + ⋯
L(s)  = 1  + 5-s − 11-s + 13-s + 17-s + 19-s − 23-s + 25-s − 29-s − 31-s − 37-s + 41-s + 43-s + 47-s − 53-s − 55-s + 61-s + 65-s + 67-s + 71-s + 73-s − 79-s + 83-s + 85-s − 89-s + 95-s + 97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4956 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4956 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4956\)    =    \(2^{2} \cdot 3 \cdot 7 \cdot 59\)
Sign: $1$
Analytic conductor: \(23.0155\)
Root analytic conductor: \(23.0155\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{4956} (4955, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 4956,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.322819100\)
\(L(\frac12)\) \(\approx\) \(2.322819100\)
\(L(1)\) \(\approx\) \(1.332663126\)
\(L(1)\) \(\approx\) \(1.332663126\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
59 \( 1 \)
good5 \( 1 + T \)
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 \( 1 - T \)
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

−18.2167438417979873361028843658, −17.48229655997999588222430896481, −16.7381759052878446282191269483, −16.00167090904608472766688697187, −15.64457447290641047368548942805, −14.48375682137062039560208792598, −14.08102637672447582124652567482, −13.44137602599828509718981837352, −12.765956496140261522047108631981, −12.203342893411006030575499446166, −11.08726140149086972655228124670, −10.7236338164853065788608011235, −9.82694396456322933301870183589, −9.438943718574286507000047618944, −8.54690831365452319362490091786, −7.79083065125069111002417036432, −7.17267462823029838798502400888, −6.14967643076948419357228722956, −5.55372978464012886430462485585, −5.22644976593356612341580570006, −3.9703572498646952514753058503, −3.304560621740529097748161905712, −2.41239128000362137782725010417, −1.69352009317977673872047032467, −0.8061223725513337145172252190, 0.8061223725513337145172252190, 1.69352009317977673872047032467, 2.41239128000362137782725010417, 3.304560621740529097748161905712, 3.9703572498646952514753058503, 5.22644976593356612341580570006, 5.55372978464012886430462485585, 6.14967643076948419357228722956, 7.17267462823029838798502400888, 7.79083065125069111002417036432, 8.54690831365452319362490091786, 9.438943718574286507000047618944, 9.82694396456322933301870183589, 10.7236338164853065788608011235, 11.08726140149086972655228124670, 12.203342893411006030575499446166, 12.765956496140261522047108631981, 13.44137602599828509718981837352, 14.08102637672447582124652567482, 14.48375682137062039560208792598, 15.64457447290641047368548942805, 16.00167090904608472766688697187, 16.7381759052878446282191269483, 17.48229655997999588222430896481, 18.2167438417979873361028843658

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