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Properties

Label 2-277200-1.1-c1-0-165
Degree $2$
Conductor $277200$
Sign $1$
Analytic cond. $2213.45$
Root an. cond. $47.0473$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7-s + 11-s + 6·13-s − 2·17-s + 8·19-s + 4·23-s − 2·29-s + 8·31-s − 6·37-s + 2·41-s + 8·43-s + 4·47-s + 49-s + 2·53-s − 12·59-s + 10·61-s − 12·67-s − 12·71-s − 10·73-s + 77-s + 8·79-s − 12·83-s − 10·89-s + 6·91-s + 14·97-s + 101-s + 103-s + ⋯
L(s)  = 1  + 0.377·7-s + 0.301·11-s + 1.66·13-s − 0.485·17-s + 1.83·19-s + 0.834·23-s − 0.371·29-s + 1.43·31-s − 0.986·37-s + 0.312·41-s + 1.21·43-s + 0.583·47-s + 1/7·49-s + 0.274·53-s − 1.56·59-s + 1.28·61-s − 1.46·67-s − 1.42·71-s − 1.17·73-s + 0.113·77-s + 0.900·79-s − 1.31·83-s − 1.05·89-s + 0.628·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 277200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(277200\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(2213.45\)
Root analytic conductor: \(47.0473\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 277200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.275340908\)
\(L(\frac12)\) \(\approx\) \(4.275340908\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
7 \( 1 - T \)
11 \( 1 - T \)
good13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−12.86404155531330, −12.17124503919824, −11.79724158216996, −11.29753970090429, −11.12708112393574, −10.35472840010882, −10.20405839893197, −9.365450321609066, −8.982164739276143, −8.719314194918845, −8.158131363849108, −7.466380738623028, −7.298434650919115, −6.615567483018665, −6.032803119166518, −5.728960211492320, −5.168284955766642, −4.448896243116907, −4.217698327129064, −3.298849779727863, −3.194317587315324, −2.410851361683800, −1.554533673068378, −1.184766304392094, −0.6185934252033536, 0.6185934252033536, 1.184766304392094, 1.554533673068378, 2.410851361683800, 3.194317587315324, 3.298849779727863, 4.217698327129064, 4.448896243116907, 5.168284955766642, 5.728960211492320, 6.032803119166518, 6.615567483018665, 7.298434650919115, 7.466380738623028, 8.158131363849108, 8.719314194918845, 8.982164739276143, 9.365450321609066, 10.20405839893197, 10.35472840010882, 11.12708112393574, 11.29753970090429, 11.79724158216996, 12.17124503919824, 12.86404155531330

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