Properties

Label 1-2348-2348.2347-r0-0-0
Degree $1$
Conductor $2348$
Sign $1$
Analytic cond. $10.9040$
Root an. cond. $10.9040$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2348 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2348 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2348\)    =    \(2^{2} \cdot 587\)
Sign: $1$
Analytic conductor: \(10.9040\)
Root analytic conductor: \(10.9040\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2348} (2347, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 2348,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7087340314\)
\(L(\frac12)\) \(\approx\) \(0.7087340314\)
\(L(1)\) \(\approx\) \(0.6254838274\)
\(L(1)\) \(\approx\) \(0.6254838274\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
587 \( 1 \)
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 \( 1 - T \)
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

−19.67333724292221522512104597643, −18.823271487742174420767963787949, −18.32105591094818071720742893996, −17.146177800772302876276569876642, −16.743297322115605415930555354776, −16.18642374634967463410625971338, −15.401351577817466511996137763329, −14.762108750652480147983471892516, −13.77548288215325778563294563199, −12.774117679785958328611294948446, −12.08229114941215463140063198330, −11.90492411289250037056081464657, −10.948328795762844351094799278279, −10.07381147378783789382134582730, −9.5357292479733762146963005813, −8.58382957226941129612702433225, −7.33373686458774408774160975251, −7.09173452195590922371763398989, −6.25858657270666129251843534880, −5.25321992522607798208751159344, −4.65757456198944199051348880795, −3.56484144730225331734199743982, −3.13908611128303646846942648589, −1.51198992553563685506780998969, −0.56077036460322639583224562614, 0.56077036460322639583224562614, 1.51198992553563685506780998969, 3.13908611128303646846942648589, 3.56484144730225331734199743982, 4.65757456198944199051348880795, 5.25321992522607798208751159344, 6.25858657270666129251843534880, 7.09173452195590922371763398989, 7.33373686458774408774160975251, 8.58382957226941129612702433225, 9.5357292479733762146963005813, 10.07381147378783789382134582730, 10.948328795762844351094799278279, 11.90492411289250037056081464657, 12.08229114941215463140063198330, 12.774117679785958328611294948446, 13.77548288215325778563294563199, 14.762108750652480147983471892516, 15.401351577817466511996137763329, 16.18642374634967463410625971338, 16.743297322115605415930555354776, 17.146177800772302876276569876642, 18.32105591094818071720742893996, 18.823271487742174420767963787949, 19.67333724292221522512104597643

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