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Properties

Label 4-266240-1.1-c1e2-0-5
Degree $4$
Conductor $266240$
Sign $1$
Analytic cond. $16.9756$
Root an. cond. $2.02981$
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  + 5-s − 2·9-s + 3·13-s + 6·17-s − 4·25-s − 4·37-s + 6·41-s − 2·45-s − 4·49-s + 20·61-s + 3·65-s + 4·73-s − 5·81-s + 6·85-s + 12·89-s − 2·97-s + 14·109-s + 18·113-s − 6·117-s + 14·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  + 0.447·5-s − 2/3·9-s + 0.832·13-s + 1.45·17-s − 4/5·25-s − 0.657·37-s + 0.937·41-s − 0.298·45-s − 4/7·49-s + 2.56·61-s + 0.372·65-s + 0.468·73-s − 5/9·81-s + 0.650·85-s + 1.27·89-s − 0.203·97-s + 1.34·109-s + 1.69·113-s − 0.554·117-s + 1.27·121-s − 0.357·125-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 266240 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 266240 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(266240\)    =    \(2^{12} \cdot 5 \cdot 13\)
Sign: $1$
Analytic conductor: \(16.9756\)
Root analytic conductor: \(2.02981\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 266240,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.913482450\)
\(L(\frac12)\) \(\approx\) \(1.913482450\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + p T^{2} ) \)
13$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 4 T + p T^{2} ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.3.a_c
7$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.7.a_e
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.11.a_ao
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.17.ag_bi
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.19.a_ac
23$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.23.a_aba
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.31.a_k
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.37.e_da
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.41.ag_de
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.43.a_aby
47$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.47.a_e
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \) 2.59.a_adu
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.61.au_io
67$C_2^2$ \( 1 - 80 T^{2} + p^{2} T^{4} \) 2.67.a_adc
71$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.71.a_ec
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.73.ae_fu
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.a_dq
83$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.83.a_e
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_ig
97$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.97.c_ek
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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

−8.711653343728661771526167528132, −8.582774074538030362222145471199, −7.982542477863279093102810509982, −7.57565612422620505156706958143, −7.03368088973961009476123874816, −6.41874567303196997911136434891, −5.91476978528373420238588158402, −5.63455953843307299387387063734, −5.13913165950800524739352916581, −4.44129279722864710491733915391, −3.62177654363765217999702011236, −3.40355050157353184905364617413, −2.53487467122914803149534420632, −1.82848271307513427697098660029, −0.863247218339281598497579737813, 0.863247218339281598497579737813, 1.82848271307513427697098660029, 2.53487467122914803149534420632, 3.40355050157353184905364617413, 3.62177654363765217999702011236, 4.44129279722864710491733915391, 5.13913165950800524739352916581, 5.63455953843307299387387063734, 5.91476978528373420238588158402, 6.41874567303196997911136434891, 7.03368088973961009476123874816, 7.57565612422620505156706958143, 7.982542477863279093102810509982, 8.582774074538030362222145471199, 8.711653343728661771526167528132

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