Properties

Label 2-25857-1.1-c1-0-13
Degree $2$
Conductor $25857$
Sign $1$
Analytic cond. $206.469$
Root an. cond. $14.3690$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s − 4·5-s − 3·7-s + 8·10-s − 4·11-s + 6·14-s − 4·16-s − 17-s − 4·19-s − 8·20-s + 8·22-s − 6·23-s + 11·25-s − 6·28-s + 2·29-s + 7·31-s + 8·32-s + 2·34-s + 12·35-s − 2·37-s + 8·38-s − 2·41-s + 11·43-s − 8·44-s + 12·46-s − 2·47-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 1.78·5-s − 1.13·7-s + 2.52·10-s − 1.20·11-s + 1.60·14-s − 16-s − 0.242·17-s − 0.917·19-s − 1.78·20-s + 1.70·22-s − 1.25·23-s + 11/5·25-s − 1.13·28-s + 0.371·29-s + 1.25·31-s + 1.41·32-s + 0.342·34-s + 2.02·35-s − 0.328·37-s + 1.29·38-s − 0.312·41-s + 1.67·43-s − 1.20·44-s + 1.76·46-s − 0.291·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25857 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25857 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25857\)    =    \(3^{2} \cdot 13^{2} \cdot 17\)
Sign: $1$
Analytic conductor: \(206.469\)
Root analytic conductor: \(14.3690\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{25857} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 25857,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
17 \( 1 + T \)
good2 \( 1 + p T + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 3 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.97982786629228, −15.60688582258288, −15.21654880682016, −14.39559766271876, −13.61974823697928, −13.01545602576671, −12.55713851015045, −11.93521355466865, −11.45715804734053, −10.80906999544535, −10.25558060362490, −10.07045347006894, −9.149130879341451, −8.650660135538694, −8.077665886825475, −7.845606391720963, −7.159300982849877, −6.661078442904420, −5.998055435563203, −4.930949424890872, −4.268596105996029, −3.767051091659232, −2.837258618350146, −2.344313890450583, −1.019930560911377, 0, 0, 1.019930560911377, 2.344313890450583, 2.837258618350146, 3.767051091659232, 4.268596105996029, 4.930949424890872, 5.998055435563203, 6.661078442904420, 7.159300982849877, 7.845606391720963, 8.077665886825475, 8.650660135538694, 9.149130879341451, 10.07045347006894, 10.25558060362490, 10.80906999544535, 11.45715804734053, 11.93521355466865, 12.55713851015045, 13.01545602576671, 13.61974823697928, 14.39559766271876, 15.21654880682016, 15.60688582258288, 15.97982786629228

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