Properties

Label 2-208725-1.1-c1-0-14
Degree $2$
Conductor $208725$
Sign $-1$
Analytic cond. $1666.67$
Root an. cond. $40.8249$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3-s + 2·4-s + 2·6-s − 5·7-s + 9-s − 2·12-s − 6·13-s + 10·14-s − 4·16-s + 17-s − 2·18-s − 2·19-s + 5·21-s + 23-s + 12·26-s − 27-s − 10·28-s + 29-s − 5·31-s + 8·32-s − 2·34-s + 2·36-s + 7·37-s + 4·38-s + 6·39-s + 7·41-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s + 4-s + 0.816·6-s − 1.88·7-s + 1/3·9-s − 0.577·12-s − 1.66·13-s + 2.67·14-s − 16-s + 0.242·17-s − 0.471·18-s − 0.458·19-s + 1.09·21-s + 0.208·23-s + 2.35·26-s − 0.192·27-s − 1.88·28-s + 0.185·29-s − 0.898·31-s + 1.41·32-s − 0.342·34-s + 1/3·36-s + 1.15·37-s + 0.648·38-s + 0.960·39-s + 1.09·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208725\)    =    \(3 \cdot 5^{2} \cdot 11^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(1666.67\)
Root analytic conductor: \(40.8249\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 208725,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 \)
11 \( 1 \)
23 \( 1 - T \)
good2 \( 1 + p T + p T^{2} \)
7 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 - T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−13.06284212792602, −12.68165504942170, −12.25288414920217, −11.92329346603056, −11.13106971259212, −10.75147340453941, −10.32412946485382, −9.814469451550098, −9.541750345402851, −9.253902604003219, −8.723526752503129, −7.980156188337239, −7.422269552730471, −7.243202899285689, −6.648326835160661, −6.200472918362076, −5.756370125496938, −4.969241692509509, −4.514459858717073, −3.830172927057493, −3.167106148656592, −2.502846871531705, −2.144977206642708, −1.150057823625039, −0.4874063002971954, 0, 0.4874063002971954, 1.150057823625039, 2.144977206642708, 2.502846871531705, 3.167106148656592, 3.830172927057493, 4.514459858717073, 4.969241692509509, 5.756370125496938, 6.200472918362076, 6.648326835160661, 7.243202899285689, 7.422269552730471, 7.980156188337239, 8.723526752503129, 9.253902604003219, 9.541750345402851, 9.814469451550098, 10.32412946485382, 10.75147340453941, 11.13106971259212, 11.92329346603056, 12.25288414920217, 12.68165504942170, 13.06284212792602

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