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Properties

Label 4-1320e2-1.1-c1e2-0-27
Degree $4$
Conductor $1742400$
Sign $1$
Analytic cond. $111.096$
Root an. cond. $3.24657$
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  − 2·3-s + 2·5-s + 3·9-s + 4·11-s − 4·15-s + 8·23-s + 3·25-s − 4·27-s − 8·31-s − 8·33-s + 4·37-s + 6·45-s − 8·47-s − 6·49-s + 12·53-s + 8·55-s + 8·59-s + 8·67-s − 16·69-s + 16·71-s − 6·75-s + 5·81-s + 4·89-s + 16·93-s + 20·97-s + 12·99-s − 8·103-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.894·5-s + 9-s + 1.20·11-s − 1.03·15-s + 1.66·23-s + 3/5·25-s − 0.769·27-s − 1.43·31-s − 1.39·33-s + 0.657·37-s + 0.894·45-s − 1.16·47-s − 6/7·49-s + 1.64·53-s + 1.07·55-s + 1.04·59-s + 0.977·67-s − 1.92·69-s + 1.89·71-s − 0.692·75-s + 5/9·81-s + 0.423·89-s + 1.65·93-s + 2.03·97-s + 1.20·99-s − 0.788·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1742400\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(111.096\)
Root analytic conductor: \(3.24657\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1742400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.066656278\)
\(L(\frac12)\) \(\approx\) \(2.066656278\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
3$C_1$ \( ( 1 + T )^{2} \)
5$C_1$ \( ( 1 - T )^{2} \)
11$C_2$ \( 1 - 4 T + p T^{2} \)
good7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.7.a_g
13$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.13.a_o
17$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.17.a_ak
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.19.a_aba
23$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) 2.23.ai_bu
29$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.29.a_g
31$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.31.i_ck
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.ae_ck
41$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.41.a_abi
43$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.43.a_ck
47$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.i_dq
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.ai_cs
61$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.61.a_ak
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.67.ai_fu
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.71.aq_hy
73$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.73.a_ak
79$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.79.a_aby
83$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.83.a_aby
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.ae_eo
97$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.97.au_iw
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.62519253774895718935152712329, −7.25002159665600914188162589571, −6.71437334568133501819174401250, −6.65058702207034604098072641266, −6.11514710822997490710036913972, −5.67161351285352187841730525346, −5.24231885406943723300303669309, −4.92213648063589071330438463956, −4.45101121137643499489357337423, −3.64965689503982669160373755503, −3.52820316318549649250191117552, −2.54481659549443094326361845004, −1.97305351011013827453703602783, −1.29659145659768067057495387651, −0.72193166495151223931367925701, 0.72193166495151223931367925701, 1.29659145659768067057495387651, 1.97305351011013827453703602783, 2.54481659549443094326361845004, 3.52820316318549649250191117552, 3.64965689503982669160373755503, 4.45101121137643499489357337423, 4.92213648063589071330438463956, 5.24231885406943723300303669309, 5.67161351285352187841730525346, 6.11514710822997490710036913972, 6.65058702207034604098072641266, 6.71437334568133501819174401250, 7.25002159665600914188162589571, 7.62519253774895718935152712329

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