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Properties

Label 4-2376e2-1.1-c1e2-0-2
Degree $4$
Conductor $5645376$
Sign $1$
Analytic cond. $359.954$
Root an. cond. $4.35573$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 5·11-s + 4·23-s − 7·25-s − 14·31-s − 12·37-s + 12·47-s − 5·49-s + 10·53-s − 10·55-s − 8·59-s − 20·67-s − 16·71-s + 12·89-s − 2·97-s + 8·103-s − 12·113-s − 8·115-s + 14·121-s + 26·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 28·155-s + ⋯
L(s)  = 1  − 0.894·5-s + 1.50·11-s + 0.834·23-s − 7/5·25-s − 2.51·31-s − 1.97·37-s + 1.75·47-s − 5/7·49-s + 1.37·53-s − 1.34·55-s − 1.04·59-s − 2.44·67-s − 1.89·71-s + 1.27·89-s − 0.203·97-s + 0.788·103-s − 1.12·113-s − 0.746·115-s + 1.27·121-s + 2.32·125-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 + 2.24·155-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5645376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5645376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5645376\)    =    \(2^{6} \cdot 3^{6} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(359.954\)
Root analytic conductor: \(4.35573\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5645376,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9504895845\)
\(L(\frac12)\) \(\approx\) \(0.9504895845\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11$C_2$ \( 1 - 5 T + p T^{2} \)
good5$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.5.c_l
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.7.a_f
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.17.a_abe
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.19.a_bi
23$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.23.ae_by
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.31.o_eh
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.37.m_eg
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.43.a_de
47$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.47.am_fa
53$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.53.ak_fb
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.59.i_fe
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.61.a_cg
67$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.67.u_ja
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.71.q_hy
73$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.73.a_fp
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.79.a_adu
83$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.83.a_bt
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_ig
97$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.97.c_hn
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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

−7.25448367711369655898353657421, −7.18606606038016321714885703629, −6.29044147324627544605416638180, −6.23138020629374792100300833919, −5.59874232278117491097239848832, −5.30114736290957773682776558825, −4.71525232702415007486392238194, −4.22684399554956914032355564142, −3.74514947909231141400939990651, −3.70938085646275379302926154934, −3.12566427060172588968254507211, −2.42324364537421860350327740670, −1.64081961055094255152301138209, −1.46136276597000756299193988555, −0.33400700903516745004242778259, 0.33400700903516745004242778259, 1.46136276597000756299193988555, 1.64081961055094255152301138209, 2.42324364537421860350327740670, 3.12566427060172588968254507211, 3.70938085646275379302926154934, 3.74514947909231141400939990651, 4.22684399554956914032355564142, 4.71525232702415007486392238194, 5.30114736290957773682776558825, 5.59874232278117491097239848832, 6.23138020629374792100300833919, 6.29044147324627544605416638180, 7.18606606038016321714885703629, 7.25448367711369655898353657421

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