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Properties

Label 4-240e2-1.1-c1e2-0-29
Degree $4$
Conductor $57600$
Sign $-1$
Analytic cond. $3.67262$
Root an. cond. $1.38434$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·5-s − 9-s − 5·13-s + 10·17-s + 4·25-s − 13·29-s − 5·37-s + 2·41-s + 3·45-s − 11·49-s − 10·53-s − 61-s + 15·65-s − 20·73-s + 81-s − 30·85-s + 3·89-s − 20·97-s − 101-s − 16·109-s + 10·113-s + 5·117-s + 4·121-s + 3·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 1.34·5-s − 1/3·9-s − 1.38·13-s + 2.42·17-s + 4/5·25-s − 2.41·29-s − 0.821·37-s + 0.312·41-s + 0.447·45-s − 1.57·49-s − 1.37·53-s − 0.128·61-s + 1.86·65-s − 2.34·73-s + 1/9·81-s − 3.25·85-s + 0.317·89-s − 2.03·97-s − 0.0995·101-s − 1.53·109-s + 0.940·113-s + 0.462·117-s + 4/11·121-s + 0.268·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(57600\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(3.67262\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 57600,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 + 3 T + p T^{2} \)
good7$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \) 2.7.a_l
11$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.11.a_ae
13$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.13.f_bg
17$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) 2.17.ak_cd
19$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.19.a_ai
23$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \) 2.23.a_abl
29$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.29.n_dq
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.31.a_n
37$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.37.f_bm
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.41.ac_cg
43$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.43.a_abe
47$C_2^2$ \( 1 - 19 T^{2} + p^{2} T^{4} \) 2.47.a_at
53$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.53.k_es
59$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.59.a_ai
61$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.61.b_co
67$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.67.a_ck
71$C_2^2$ \( 1 - 137 T^{2} + p^{2} T^{4} \) 2.71.a_afh
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.73.u_jm
79$C_2^2$ \( 1 + 69 T^{2} + p^{2} T^{4} \) 2.79.a_cr
83$C_2^2$ \( 1 + 60 T^{2} + p^{2} T^{4} \) 2.83.a_ci
89$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.89.ad_y
97$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.97.u_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

−9.633856258612352948799981920451, −9.413950095898854559625719759590, −8.616055116496221828080002977134, −7.955698828669111876069728402253, −7.72295349118974320635435756172, −7.36461979270040181614265986009, −6.79018737764365042759819087546, −5.80251884957167446450534139804, −5.45195519943668363961343711838, −4.80485489526250567688885927655, −4.07550605155476125475085699965, −3.38792423368953983467273395287, −2.97484260424592753553088949917, −1.65794635139681958335660676269, 0, 1.65794635139681958335660676269, 2.97484260424592753553088949917, 3.38792423368953983467273395287, 4.07550605155476125475085699965, 4.80485489526250567688885927655, 5.45195519943668363961343711838, 5.80251884957167446450534139804, 6.79018737764365042759819087546, 7.36461979270040181614265986009, 7.72295349118974320635435756172, 7.955698828669111876069728402253, 8.616055116496221828080002977134, 9.413950095898854559625719759590, 9.633856258612352948799981920451

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