Properties

Label 2-560-35.34-c4-0-74
Degree $2$
Conductor $560$
Sign $1$
Analytic cond. $57.8871$
Root an. cond. $7.60836$
Motivic weight $4$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 17·3-s + 25·5-s − 49·7-s + 208·9-s + 73·11-s + 23·13-s + 425·15-s + 263·17-s − 833·21-s + 625·25-s + 2.15e3·27-s − 1.15e3·29-s + 1.24e3·33-s − 1.22e3·35-s + 391·39-s + 5.20e3·45-s + 3.45e3·47-s + 2.40e3·49-s + 4.47e3·51-s + 1.82e3·55-s − 1.01e4·63-s + 575·65-s + 1.00e4·71-s − 9.50e3·73-s + 1.06e4·75-s − 3.57e3·77-s − 1.21e4·79-s + ⋯
L(s)  = 1  + 17/9·3-s + 5-s − 7-s + 2.56·9-s + 0.603·11-s + 0.136·13-s + 17/9·15-s + 0.910·17-s − 1.88·21-s + 25-s + 2.96·27-s − 1.37·29-s + 1.13·33-s − 35-s + 0.257·39-s + 2.56·45-s + 1.56·47-s + 49-s + 1.71·51-s + 0.603·55-s − 2.56·63-s + 0.136·65-s + 1.99·71-s − 1.78·73-s + 17/9·75-s − 0.603·77-s − 1.94·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $1$
Analytic conductor: \(57.8871\)
Root analytic conductor: \(7.60836\)
Motivic weight: \(4\)
Rational: yes
Arithmetic: yes
Character: $\chi_{560} (209, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :2),\ 1)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(5.252136353\)
\(L(\frac12)\) \(\approx\) \(5.252136353\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - p^{2} T \)
7 \( 1 + p^{2} T \)
good3 \( 1 - 17 T + p^{4} T^{2} \)
11 \( 1 - 73 T + p^{4} T^{2} \)
13 \( 1 - 23 T + p^{4} T^{2} \)
17 \( 1 - 263 T + p^{4} T^{2} \)
19 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
23 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
29 \( 1 + 1153 T + p^{4} T^{2} \)
31 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
37 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
41 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
43 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
47 \( 1 - 3457 T + p^{4} T^{2} \)
53 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
59 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
61 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
67 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
71 \( 1 - 10078 T + p^{4} T^{2} \)
73 \( 1 + 9502 T + p^{4} T^{2} \)
79 \( 1 + 12167 T + p^{4} T^{2} \)
83 \( 1 - 6382 T + p^{4} T^{2} \)
89 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
97 \( 1 - 3383 T + p^{4} T^{2} \)
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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

−9.756140813591320034441824980498, −9.336646253839553075558082406766, −8.651640066023227497976634278692, −7.54515744246965041150994945580, −6.75482690601243691295376509576, −5.65193898697282704457783897797, −4.06136872815027826709992524710, −3.24026241563139815315029036209, −2.34696017130252827452066148699, −1.27203549294382118856399719008, 1.27203549294382118856399719008, 2.34696017130252827452066148699, 3.24026241563139815315029036209, 4.06136872815027826709992524710, 5.65193898697282704457783897797, 6.75482690601243691295376509576, 7.54515744246965041150994945580, 8.651640066023227497976634278692, 9.336646253839553075558082406766, 9.756140813591320034441824980498

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