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Properties

Label 4-2376e2-1.1-c1e2-0-8
Degree $4$
Conductor $5645376$
Sign $-1$
Analytic cond. $359.954$
Root an. cond. $4.35573$
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  − 2·5-s − 4·11-s + 12·23-s − 6·25-s + 2·31-s − 10·37-s − 4·47-s + 49-s − 20·53-s + 8·55-s + 4·59-s − 6·67-s − 12·71-s + 16·89-s + 14·97-s + 22·103-s + 22·113-s − 24·115-s + 5·121-s + 22·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 4·155-s + ⋯
L(s)  = 1  − 0.894·5-s − 1.20·11-s + 2.50·23-s − 6/5·25-s + 0.359·31-s − 1.64·37-s − 0.583·47-s + 1/7·49-s − 2.74·53-s + 1.07·55-s + 0.520·59-s − 0.733·67-s − 1.42·71-s + 1.69·89-s + 1.42·97-s + 2.16·103-s + 2.06·113-s − 2.23·115-s + 5/11·121-s + 1.96·125-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 − 0.321·155-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: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 5645376,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
11$C_2$ \( 1 + 4 T + p T^{2} \)
good5$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.c_k
7$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.7.a_ab
13$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.13.a_s
17$C_2^2$ \( 1 - 31 T^{2} + p^{2} T^{4} \) 2.17.a_abf
19$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.19.a_g
23$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) 2.23.am_cv
29$C_2^2$ \( 1 + 41 T^{2} + p^{2} T^{4} \) 2.29.a_bp
31$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.31.ac_cc
37$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.37.k_df
41$C_2^2$ \( 1 + 37 T^{2} + p^{2} T^{4} \) 2.41.a_bl
43$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \) 2.43.a_an
47$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.47.e_dt
53$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.53.u_hy
59$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.59.ae_dt
61$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.61.a_ba
67$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.67.g_eo
71$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.71.m_gs
73$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \) 2.73.a_cw
79$C_2^2$ \( 1 + 75 T^{2} + p^{2} T^{4} \) 2.79.a_cx
83$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \) 2.83.a_ady
89$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.89.aq_fm
97$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.97.ao_jj
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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

−7.32994653682455203611902037691, −6.66938418564011166987025652168, −6.27715350162813254254588582572, −5.88651630357657035997218510968, −5.29201046300854552098981090022, −4.89838737211202977970256624193, −4.75786338418925890570893868626, −4.18963825651619011324814561123, −3.48495088242546355656873096872, −3.22580159155146888967667912543, −2.92849382979062638668894882017, −2.11186382487233464040819101983, −1.64833873661739608707906833916, −0.73162045413681380754587187591, 0, 0.73162045413681380754587187591, 1.64833873661739608707906833916, 2.11186382487233464040819101983, 2.92849382979062638668894882017, 3.22580159155146888967667912543, 3.48495088242546355656873096872, 4.18963825651619011324814561123, 4.75786338418925890570893868626, 4.89838737211202977970256624193, 5.29201046300854552098981090022, 5.88651630357657035997218510968, 6.27715350162813254254588582572, 6.66938418564011166987025652168, 7.32994653682455203611902037691

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