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Properties

Label 2-20-20.19-c2-0-2
Degree $2$
Conductor $20$
Sign $1$
Analytic cond. $0.544960$
Root an. cond. $0.738214$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 4·3-s + 4·4-s − 5·5-s − 8·6-s + 4·7-s + 8·8-s + 7·9-s − 10·10-s − 16·12-s + 8·14-s + 20·15-s + 16·16-s + 14·18-s − 20·20-s − 16·21-s − 44·23-s − 32·24-s + 25·25-s + 8·27-s + 16·28-s − 22·29-s + 40·30-s + 32·32-s − 20·35-s + 28·36-s − 40·40-s + ⋯
L(s)  = 1  + 2-s − 4/3·3-s + 4-s − 5-s − 4/3·6-s + 4/7·7-s + 8-s + 7/9·9-s − 10-s − 4/3·12-s + 4/7·14-s + 4/3·15-s + 16-s + 7/9·18-s − 20-s − 0.761·21-s − 1.91·23-s − 4/3·24-s + 25-s + 8/27·27-s + 4/7·28-s − 0.758·29-s + 4/3·30-s + 32-s − 4/7·35-s + 7/9·36-s − 40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $1$
Analytic conductor: \(0.544960\)
Root analytic conductor: \(0.738214\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: $\chi_{20} (19, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9787967191\)
\(L(\frac12)\) \(\approx\) \(0.9787967191\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p T \)
5 \( 1 + p T \)
good3 \( 1 + 4 T + p^{2} T^{2} \)
7 \( 1 - 4 T + p^{2} T^{2} \)
11 \( ( 1 - p T )( 1 + p T ) \)
13 \( ( 1 - p T )( 1 + p T ) \)
17 \( ( 1 - p T )( 1 + p T ) \)
19 \( ( 1 - p T )( 1 + p T ) \)
23 \( 1 + 44 T + p^{2} T^{2} \)
29 \( 1 + 22 T + p^{2} T^{2} \)
31 \( ( 1 - p T )( 1 + p T ) \)
37 \( ( 1 - p T )( 1 + p T ) \)
41 \( 1 - 62 T + p^{2} T^{2} \)
43 \( 1 - 76 T + p^{2} T^{2} \)
47 \( 1 - 4 T + p^{2} T^{2} \)
53 \( ( 1 - p T )( 1 + p T ) \)
59 \( ( 1 - p T )( 1 + p T ) \)
61 \( 1 + 58 T + p^{2} T^{2} \)
67 \( 1 + 116 T + p^{2} T^{2} \)
71 \( ( 1 - p T )( 1 + p T ) \)
73 \( ( 1 - p T )( 1 + p T ) \)
79 \( ( 1 - p T )( 1 + p T ) \)
83 \( 1 - 76 T + p^{2} T^{2} \)
89 \( 1 + 142 T + p^{2} T^{2} \)
97 \( ( 1 - p T )( 1 + p T ) \)
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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

−17.94623264217154549029963326853, −16.58069703249018250095573939708, −15.71356859000119229857085211081, −14.33799382156206816648283419331, −12.50616367880551875248546168406, −11.64696205708223212946194753064, −10.76146029902745408520576312678, −7.63461388081077863452249905247, −5.92589782679196628096697547408, −4.36926853379325160824457216794, 4.36926853379325160824457216794, 5.92589782679196628096697547408, 7.63461388081077863452249905247, 10.76146029902745408520576312678, 11.64696205708223212946194753064, 12.50616367880551875248546168406, 14.33799382156206816648283419331, 15.71356859000119229857085211081, 16.58069703249018250095573939708, 17.94623264217154549029963326853

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