Properties

Label 2-1526-1.1-c1-0-50
Degree $2$
Conductor $1526$
Sign $-1$
Analytic cond. $12.1851$
Root an. cond. $3.49072$
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-s − 3-s + 4-s + 5-s − 6-s + 7-s + 8-s − 2·9-s + 10-s − 3·11-s − 12-s − 6·13-s + 14-s − 15-s + 16-s − 2·17-s − 2·18-s + 20-s − 21-s − 3·22-s − 6·23-s − 24-s − 4·25-s − 6·26-s + 5·27-s + 28-s − 30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s − 2/3·9-s + 0.316·10-s − 0.904·11-s − 0.288·12-s − 1.66·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.471·18-s + 0.223·20-s − 0.218·21-s − 0.639·22-s − 1.25·23-s − 0.204·24-s − 4/5·25-s − 1.17·26-s + 0.962·27-s + 0.188·28-s − 0.182·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1526 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1526 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1526\)    =    \(2 \cdot 7 \cdot 109\)
Sign: $-1$
Analytic conductor: \(12.1851\)
Root analytic conductor: \(3.49072\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1526,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 - T \)
7 \( 1 - T \)
109 \( 1 + T \)
good3 \( 1 + T + p T^{2} \)
5 \( 1 - T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−9.117828627426357441137464936094, −8.065281149853209060616012078115, −7.37254497052080588577371673056, −6.39940323385614452260114207042, −5.54350958210853846360207828376, −5.11031490968013398287456926487, −4.17175621228422883644320076210, −2.76243985087989233558008934289, −2.06208988293120625162340121374, 0, 2.06208988293120625162340121374, 2.76243985087989233558008934289, 4.17175621228422883644320076210, 5.11031490968013398287456926487, 5.54350958210853846360207828376, 6.39940323385614452260114207042, 7.37254497052080588577371673056, 8.065281149853209060616012078115, 9.117828627426357441137464936094

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