Properties

Label 2-26520-1.1-c1-0-14
Degree $2$
Conductor $26520$
Sign $-1$
Analytic cond. $211.763$
Root an. cond. $14.5520$
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  − 3-s − 5-s + 2·7-s + 9-s + 13-s + 15-s − 17-s − 2·21-s + 25-s − 27-s − 2·29-s + 2·31-s − 2·35-s + 8·37-s − 39-s + 2·41-s − 4·43-s − 45-s − 6·47-s − 3·49-s + 51-s + 2·53-s + 2·59-s − 10·61-s + 2·63-s − 65-s + 4·67-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.277·13-s + 0.258·15-s − 0.242·17-s − 0.436·21-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.359·31-s − 0.338·35-s + 1.31·37-s − 0.160·39-s + 0.312·41-s − 0.609·43-s − 0.149·45-s − 0.875·47-s − 3/7·49-s + 0.140·51-s + 0.274·53-s + 0.260·59-s − 1.28·61-s + 0.251·63-s − 0.124·65-s + 0.488·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(26520\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 13 \cdot 17\)
Sign: $-1$
Analytic conductor: \(211.763\)
Root analytic conductor: \(14.5520\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{26520} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 26520,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 + T \)
5 \( 1 + T \)
13 \( 1 - T \)
17 \( 1 + T \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
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

−15.51100339625109, −15.09990195205043, −14.53063033129549, −14.06762571680509, −13.18086452869658, −13.03763832254445, −12.17147863615974, −11.75344889555165, −11.20466022379755, −10.94373193158637, −10.18278216263964, −9.655618303852058, −8.923694830629873, −8.355545016926837, −7.774984610448705, −7.329508271816564, −6.513272197381213, −6.079733882347197, −5.301574124625093, −4.727048611884678, −4.236353067185309, −3.490495687505243, −2.656674469415320, −1.750775518372726, −1.026283271148462, 0, 1.026283271148462, 1.750775518372726, 2.656674469415320, 3.490495687505243, 4.236353067185309, 4.727048611884678, 5.301574124625093, 6.079733882347197, 6.513272197381213, 7.329508271816564, 7.774984610448705, 8.355545016926837, 8.923694830629873, 9.655618303852058, 10.18278216263964, 10.94373193158637, 11.20466022379755, 11.75344889555165, 12.17147863615974, 13.03763832254445, 13.18086452869658, 14.06762571680509, 14.53063033129549, 15.09990195205043, 15.51100339625109

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