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Properties

Label 4-1320e2-1.1-c1e2-0-42
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 + 9-s + 6·11-s + 4·15-s + 2·23-s − 25-s − 4·27-s − 8·31-s + 12·33-s + 4·37-s + 2·45-s + 2·47-s − 6·49-s + 4·53-s + 12·55-s + 16·59-s + 10·67-s + 4·69-s + 4·71-s − 2·75-s − 11·81-s + 32·89-s − 16·93-s − 12·97-s + 6·99-s − 22·103-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.894·5-s + 1/3·9-s + 1.80·11-s + 1.03·15-s + 0.417·23-s − 1/5·25-s − 0.769·27-s − 1.43·31-s + 2.08·33-s + 0.657·37-s + 0.298·45-s + 0.291·47-s − 6/7·49-s + 0.549·53-s + 1.61·55-s + 2.08·59-s + 1.22·67-s + 0.481·69-s + 0.474·71-s − 0.230·75-s − 1.22·81-s + 3.39·89-s − 1.65·93-s − 1.21·97-s + 0.603·99-s − 2.16·103-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

\(L(1)\) \(\approx\) \(4.705849741\)
\(L(\frac12)\) \(\approx\) \(4.705849741\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 2 T + p T^{2} \)
5$C_2$ \( 1 - 2 T + p T^{2} \)
11$C_2$ \( 1 - 6 T + p T^{2} \)
good7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.7.a_g
13$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.13.a_c
17$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.17.a_w
19$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.19.a_g
23$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.ac_w
29$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.29.a_ag
31$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.31.i_cw
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.41.a_ck
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.a_w
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.47.ac_bu
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.53.ae_dq
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.59.aq_gk
61$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \) 2.61.a_acc
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.67.ak_eg
71$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.71.ae_fm
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.73.a_abu
79$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.79.a_ade
83$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \) 2.83.a_aeo
89$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 - 14 T + p T^{2} ) \) 2.89.abg_qo
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.m_gk
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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

−7.992702249212472657533728301942, −7.35274570434748825354418995695, −6.96902331401254767443878249953, −6.63266076744712491907513518911, −6.16950901814084364897872247632, −5.60271339174011198954284276409, −5.41154997542091708326734134987, −4.63043939344072261016224231478, −4.11498963178225490542874836768, −3.60182667398516046267030120898, −3.42260147080236537370067647492, −2.57426604435551614905342585177, −2.10457637657166204437880532536, −1.67785896414465448623758569529, −0.870780218679902862613190774276, 0.870780218679902862613190774276, 1.67785896414465448623758569529, 2.10457637657166204437880532536, 2.57426604435551614905342585177, 3.42260147080236537370067647492, 3.60182667398516046267030120898, 4.11498963178225490542874836768, 4.63043939344072261016224231478, 5.41154997542091708326734134987, 5.60271339174011198954284276409, 6.16950901814084364897872247632, 6.63266076744712491907513518911, 6.96902331401254767443878249953, 7.35274570434748825354418995695, 7.992702249212472657533728301942

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