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Properties

Label 4-72e3-1.1-c1e2-0-20
Degree $4$
Conductor $373248$
Sign $-1$
Analytic cond. $23.7986$
Root an. cond. $2.20870$
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  − 2·13-s − 8·23-s + 2·25-s + 2·37-s − 8·47-s − 49-s + 2·61-s + 16·71-s − 6·73-s − 16·83-s − 6·97-s − 16·107-s + 12·109-s − 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 19·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 0.554·13-s − 1.66·23-s + 2/5·25-s + 0.328·37-s − 1.16·47-s − 1/7·49-s + 0.256·61-s + 1.89·71-s − 0.702·73-s − 1.75·83-s − 0.609·97-s − 1.54·107-s + 1.14·109-s − 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.46·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373248 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373248 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(373248\)    =    \(2^{9} \cdot 3^{6}\)
Sign: $-1$
Analytic conductor: \(23.7986\)
Root analytic conductor: \(2.20870\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 373248,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
7$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.7.a_b
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.11.a_s
13$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.13.c_x
17$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.17.a_ak
19$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \) 2.19.a_f
23$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.i_cg
29$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \) 2.29.a_abm
31$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.31.a_abi
37$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.37.ac_ct
41$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.41.a_abe
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$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.47.i_ec
53$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \) 2.53.a_acc
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.59.a_de
61$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.61.ac_dj
67$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \) 2.67.a_ad
71$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + p T^{2} ) \) 2.71.aq_fm
73$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.73.g_dn
79$C_2^2$ \( 1 + 65 T^{2} + p^{2} T^{4} \) 2.79.a_cn
83$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.q_ig
89$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.89.a_acg
97$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.97.g_hf
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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

−8.352431509173052435555840185299, −8.017495690621158276147905660496, −7.66904806947982597090713034703, −6.97982181306480047165065335560, −6.68429233484490626952056621788, −6.09863946529523077632152584819, −5.64067682367067562482091701438, −5.09943682224705208760613508941, −4.57270145412342547366264869625, −4.02178856133546181592510380165, −3.49382481358562834607248388200, −2.72839452354597333286176939418, −2.17551363884229321652769590106, −1.33628557789555324912287167886, 0, 1.33628557789555324912287167886, 2.17551363884229321652769590106, 2.72839452354597333286176939418, 3.49382481358562834607248388200, 4.02178856133546181592510380165, 4.57270145412342547366264869625, 5.09943682224705208760613508941, 5.64067682367067562482091701438, 6.09863946529523077632152584819, 6.68429233484490626952056621788, 6.97982181306480047165065335560, 7.66904806947982597090713034703, 8.017495690621158276147905660496, 8.352431509173052435555840185299

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