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Properties

Label 1-2347-2347.2346-r1-0-0
Degree $1$
Conductor $2347$
Sign $1$
Analytic cond. $252.220$
Root an. cond. $252.220$
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  − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s − 11-s − 12-s − 13-s + 14-s + 15-s + 16-s + 17-s − 18-s + 19-s − 20-s + 21-s + 22-s − 23-s + 24-s + 25-s + 26-s − 27-s − 28-s + ⋯
L(s)  = 1  − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s − 11-s − 12-s − 13-s + 14-s + 15-s + 16-s + 17-s − 18-s + 19-s − 20-s + 21-s + 22-s − 23-s + 24-s + 25-s + 26-s − 27-s − 28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2347 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2347 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2347\)
Sign: $1$
Analytic conductor: \(252.220\)
Root analytic conductor: \(252.220\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2347} (2346, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 2347,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2241703546\)
\(L(\frac12)\) \(\approx\) \(0.2241703546\)
\(L(1)\) \(\approx\) \(0.3242375672\)
\(L(1)\) \(\approx\) \(0.3242375672\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2347 \( 1 \)
good2 \( 1 - T \)
3 \( 1 - T \)
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

−19.26238750951747940034525118707, −18.53110978101683251773073702989, −18.23590322554603183135405817374, −17.12273018988534956309414500531, −16.56890187869767345439327146349, −16.028519140662761242372638581591, −15.53675479497193362699825749100, −14.77659877707709609220974026438, −13.40820783294972308459273554587, −12.30867939448506942221764431040, −12.22445208367164802434921492637, −11.37172866868655056121288873551, −10.54423426579633705314149606150, −9.92529349670977574585001617144, −9.43622168603709345085744298502, −8.11247817528232230190539988955, −7.50206425817934665161547728802, −7.08874974376451551440628365982, −6.00284316773688143096675347039, −5.43418950141067023410110458206, −4.30285950389929486487855396059, −3.272941065950403982466572107840, −2.50940337081810409979670948162, −1.10697331693776989043860215758, −0.269873707217955706046646639232, 0.269873707217955706046646639232, 1.10697331693776989043860215758, 2.50940337081810409979670948162, 3.272941065950403982466572107840, 4.30285950389929486487855396059, 5.43418950141067023410110458206, 6.00284316773688143096675347039, 7.08874974376451551440628365982, 7.50206425817934665161547728802, 8.11247817528232230190539988955, 9.43622168603709345085744298502, 9.92529349670977574585001617144, 10.54423426579633705314149606150, 11.37172866868655056121288873551, 12.22445208367164802434921492637, 12.30867939448506942221764431040, 13.40820783294972308459273554587, 14.77659877707709609220974026438, 15.53675479497193362699825749100, 16.028519140662761242372638581591, 16.56890187869767345439327146349, 17.12273018988534956309414500531, 18.23590322554603183135405817374, 18.53110978101683251773073702989, 19.26238750951747940034525118707

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