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Properties

Label 1-5653-5653.5652-r0-0-0
Degree $1$
Conductor $5653$
Sign $1$
Analytic cond. $26.2524$
Root an. cond. $26.2524$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 5653 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5653 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5653\)
Sign: $1$
Analytic conductor: \(26.2524\)
Root analytic conductor: \(26.2524\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{5653} (5652, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 5653,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.263468487\)
\(L(\frac12)\) \(\approx\) \(1.263468487\)
\(L(1)\) \(\approx\) \(0.8527415348\)
\(L(1)\) \(\approx\) \(0.8527415348\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5653 \( 1 \)
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

−18.0223988585714085459480198808, −17.154530796331810892774141573234, −16.53069938638492872254902314676, −15.84459457934830088741497124413, −14.992366307812447574061365131972, −14.90151630582620094674355095574, −14.250952403151269002889171485422, −12.89439023898246207368247681027, −12.61099703381142142129367406410, −11.69523622468001428402432830486, −11.00385532739989029591292045623, −10.46611287729553514453485159786, −9.710589794846955097929349868558, −8.93902782432096164140728382715, −8.286167365386579018815648396755, −7.85045539170241740219362034163, −7.37101100942176392120384505618, −6.798711795937275231556944552603, −5.37055007711959511254906659542, −4.830089999764738825736646683480, −3.761464675074179404948175707243, −3.12422037418501272561331931807, −2.286736489740724936320953076029, −1.707833914340464231549803287691, −0.61239061440429241741978256110, 0.61239061440429241741978256110, 1.707833914340464231549803287691, 2.286736489740724936320953076029, 3.12422037418501272561331931807, 3.761464675074179404948175707243, 4.830089999764738825736646683480, 5.37055007711959511254906659542, 6.798711795937275231556944552603, 7.37101100942176392120384505618, 7.85045539170241740219362034163, 8.286167365386579018815648396755, 8.93902782432096164140728382715, 9.710589794846955097929349868558, 10.46611287729553514453485159786, 11.00385532739989029591292045623, 11.69523622468001428402432830486, 12.61099703381142142129367406410, 12.89439023898246207368247681027, 14.250952403151269002889171485422, 14.90151630582620094674355095574, 14.992366307812447574061365131972, 15.84459457934830088741497124413, 16.53069938638492872254902314676, 17.154530796331810892774141573234, 18.0223988585714085459480198808

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