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Properties

Label 1-2408-2408.1203-r1-0-0
Degree $1$
Conductor $2408$
Sign $1$
Analytic cond. $258.775$
Root an. cond. $258.775$
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 − 45-s + 47-s − 51-s − 53-s − 55-s + 57-s − 59-s − 61-s − 65-s + 67-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 − 45-s + 47-s − 51-s − 53-s − 55-s + 57-s − 59-s − 61-s − 65-s + 67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2408 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2408 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2408\)    =    \(2^{3} \cdot 7 \cdot 43\)
Sign: $1$
Analytic conductor: \(258.775\)
Root analytic conductor: \(258.775\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2408} (1203, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 2408,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.595361012\)
\(L(\frac12)\) \(\approx\) \(3.595361012\)
\(L(1)\) \(\approx\) \(1.536501091\)
\(L(1)\) \(\approx\) \(1.536501091\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
43 \( 1 \)
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 \( 1 - T \)
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

−19.59752048769764271029942477988, −18.66235862860437042271940620362, −18.24192848613403921302563245346, −17.21028094516558279930615175198, −16.2084451601393311866427974542, −15.625088474577238823279731621333, −15.27062308457790869264248455209, −14.07672886626330624357160774832, −13.939155443342969488078138581753, −12.917695058632605209875218743855, −12.077258455254393441675292660174, −11.49411689091696350064837159693, −10.63689613671007357788372744089, −9.69006693241800146659706265180, −8.95811912733072744790102968539, −8.31851253843278691864834317081, −7.754061778904878299200158983536, −6.80537072609181090683349098212, −6.23519165664579009413958913363, −4.74764376454267636947763147528, −4.12479860667366847696413907, −3.49317080472682273291210837702, −2.705842263988754807613485239173, −1.555641721283170777074671591861, −0.74965021086765397710300950096, 0.74965021086765397710300950096, 1.555641721283170777074671591861, 2.705842263988754807613485239173, 3.49317080472682273291210837702, 4.12479860667366847696413907, 4.74764376454267636947763147528, 6.23519165664579009413958913363, 6.80537072609181090683349098212, 7.754061778904878299200158983536, 8.31851253843278691864834317081, 8.95811912733072744790102968539, 9.69006693241800146659706265180, 10.63689613671007357788372744089, 11.49411689091696350064837159693, 12.077258455254393441675292660174, 12.917695058632605209875218743855, 13.939155443342969488078138581753, 14.07672886626330624357160774832, 15.27062308457790869264248455209, 15.625088474577238823279731621333, 16.2084451601393311866427974542, 17.21028094516558279930615175198, 18.24192848613403921302563245346, 18.66235862860437042271940620362, 19.59752048769764271029942477988

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