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Properties

Label 2-223440-1.1-c1-0-109
Degree $2$
Conductor $223440$
Sign $1$
Analytic cond. $1784.17$
Root an. cond. $42.2395$
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  + 3-s + 5-s + 9-s + 2·13-s + 15-s + 2·17-s − 19-s + 4·23-s + 25-s + 27-s − 2·29-s + 8·31-s + 6·37-s + 2·39-s − 6·41-s + 4·43-s + 45-s + 2·51-s − 2·53-s − 57-s + 4·59-s + 10·61-s + 2·65-s + 4·67-s + 4·69-s + 4·71-s − 2·73-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.554·13-s + 0.258·15-s + 0.485·17-s − 0.229·19-s + 0.834·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 1.43·31-s + 0.986·37-s + 0.320·39-s − 0.937·41-s + 0.609·43-s + 0.149·45-s + 0.280·51-s − 0.274·53-s − 0.132·57-s + 0.520·59-s + 1.28·61-s + 0.248·65-s + 0.488·67-s + 0.481·69-s + 0.474·71-s − 0.234·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 223440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 223440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(223440\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(1784.17\)
Root analytic conductor: \(42.2395\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 223440,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.287801546\)
\(L(\frac12)\) \(\approx\) \(5.287801546\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 \)
19 \( 1 + T \)
good11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
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 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−12.99576929522435, −12.72198744765371, −11.95996034271805, −11.62977775651913, −11.05006777179631, −10.57354004994706, −10.06932915080242, −9.721116487823250, −9.165416478234569, −8.731139940888571, −8.321117108181253, −7.741654491900061, −7.396125753993798, −6.556158232112728, −6.425293029899915, −5.782324434358826, −5.068506109782054, −4.815038822568828, −3.990264701050248, −3.589236864047773, −2.986756812499418, −2.422382717954141, −1.922714781776360, −1.098695040664227, −0.6916471947261522, 0.6916471947261522, 1.098695040664227, 1.922714781776360, 2.422382717954141, 2.986756812499418, 3.589236864047773, 3.990264701050248, 4.815038822568828, 5.068506109782054, 5.782324434358826, 6.425293029899915, 6.556158232112728, 7.396125753993798, 7.741654491900061, 8.321117108181253, 8.731139940888571, 9.165416478234569, 9.721116487823250, 10.06932915080242, 10.57354004994706, 11.05006777179631, 11.62977775651913, 11.95996034271805, 12.72198744765371, 12.99576929522435

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