Properties

Label 2-26520-1.1-c1-0-16
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 − 5·11-s − 13-s − 15-s + 17-s − 7·19-s + 2·21-s + 4·23-s + 25-s + 27-s − 4·29-s + 3·31-s − 5·33-s − 2·35-s + 11·37-s − 39-s + 4·41-s − 43-s − 45-s + 6·47-s − 3·49-s + 51-s + 10·53-s + 5·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s − 1.50·11-s − 0.277·13-s − 0.258·15-s + 0.242·17-s − 1.60·19-s + 0.436·21-s + 0.834·23-s + 1/5·25-s + 0.192·27-s − 0.742·29-s + 0.538·31-s − 0.870·33-s − 0.338·35-s + 1.80·37-s − 0.160·39-s + 0.624·41-s − 0.152·43-s − 0.149·45-s + 0.875·47-s − 3/7·49-s + 0.140·51-s + 1.37·53-s + 0.674·55-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 \( 1 + 5 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−15.27142658181793, −15.02865412708012, −14.77289624884107, −13.91009265000339, −13.42557622939272, −12.90729003886077, −12.47714858089646, −11.84204074028480, −11.05100398274593, −10.72186726851530, −10.32260503640825, −9.386677134598714, −9.008020698977670, −8.225686815813482, −7.804989276021833, −7.591609108028988, −6.705621617054243, −5.991622607679600, −5.259618162163796, −4.598727936841616, −4.243124450271485, −3.290042580885291, −2.574315160270319, −2.139105685161413, −1.064314545002271, 0, 1.064314545002271, 2.139105685161413, 2.574315160270319, 3.290042580885291, 4.243124450271485, 4.598727936841616, 5.259618162163796, 5.991622607679600, 6.705621617054243, 7.591609108028988, 7.804989276021833, 8.225686815813482, 9.008020698977670, 9.386677134598714, 10.32260503640825, 10.72186726851530, 11.05100398274593, 11.84204074028480, 12.47714858089646, 12.90729003886077, 13.42557622939272, 13.91009265000339, 14.77289624884107, 15.02865412708012, 15.27142658181793

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