Properties

Label 2-280-280.69-c2-0-26
Degree $2$
Conductor $280$
Sign $1$
Analytic cond. $7.62944$
Root an. cond. $2.76214$
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 + 4·4-s − 5·5-s − 7·7-s − 8·8-s + 9·9-s + 10·10-s + 14·14-s + 16·16-s + 6·17-s − 18·18-s + 18·19-s − 20·20-s + 25·25-s − 28·28-s − 32·32-s − 12·34-s + 35·35-s + 36·36-s + 66·37-s − 36·38-s + 40·40-s + 54·43-s − 45·45-s + 66·47-s + 49·49-s − 50·50-s + ⋯
L(s)  = 1  − 2-s + 4-s − 5-s − 7-s − 8-s + 9-s + 10-s + 14-s + 16-s + 6/17·17-s − 18-s + 0.947·19-s − 20-s + 25-s − 28-s − 32-s − 0.352·34-s + 35-s + 36-s + 1.78·37-s − 0.947·38-s + 40-s + 1.25·43-s − 45-s + 1.40·47-s + 49-s − 50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $1$
Analytic conductor: \(7.62944\)
Root analytic conductor: \(2.76214\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: $\chi_{280} (69, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7960716764\)
\(L(\frac12)\) \(\approx\) \(0.7960716764\)
\(L(2)\) 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 \( 1 + p T \)
7 \( 1 + p T \)
good3 \( ( 1 - p T )( 1 + p T ) \)
11 \( ( 1 - p T )( 1 + p T ) \)
13 \( ( 1 - p T )( 1 + p T ) \)
17 \( 1 - 6 T + p^{2} T^{2} \)
19 \( 1 - 18 T + p^{2} T^{2} \)
23 \( ( 1 - p T )( 1 + p T ) \)
29 \( ( 1 - p T )( 1 + p T ) \)
31 \( ( 1 - p T )( 1 + p T ) \)
37 \( 1 - 66 T + p^{2} T^{2} \)
41 \( ( 1 - p T )( 1 + p T ) \)
43 \( 1 - 54 T + p^{2} T^{2} \)
47 \( 1 - 66 T + p^{2} T^{2} \)
53 \( 1 - 34 T + p^{2} T^{2} \)
59 \( 1 + 62 T + p^{2} T^{2} \)
61 \( 1 - 102 T + p^{2} T^{2} \)
67 \( 1 - 6 T + p^{2} T^{2} \)
71 \( 1 + 138 T + p^{2} T^{2} \)
73 \( 1 + 106 T + p^{2} T^{2} \)
79 \( 1 + 122 T + p^{2} T^{2} \)
83 \( ( 1 - p T )( 1 + p T ) \)
89 \( ( 1 - p T )( 1 + p T ) \)
97 \( 1 - 166 T + p^{2} T^{2} \)
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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

−11.56894588447602832468052495251, −10.50045757818459696598644216440, −9.732804369326469113273753727880, −8.868839311487043171298874818654, −7.59777285474744674749979553174, −7.15968457370650158325256000374, −5.92557191221327684887278827519, −4.10818313853348767961094869731, −2.89665292682466055369779614696, −0.864877745874781233658464391018, 0.864877745874781233658464391018, 2.89665292682466055369779614696, 4.10818313853348767961094869731, 5.92557191221327684887278827519, 7.15968457370650158325256000374, 7.59777285474744674749979553174, 8.868839311487043171298874818654, 9.732804369326469113273753727880, 10.50045757818459696598644216440, 11.56894588447602832468052495251

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