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Properties

Label 4-2376e2-1.1-c1e2-0-17
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  − 3-s + 5·5-s + 9-s + 11-s − 5·15-s − 7·23-s + 11·25-s − 27-s − 17·31-s − 33-s + 18·37-s + 5·45-s + 6·47-s + 4·49-s − 7·53-s + 5·55-s − 7·59-s − 5·67-s + 7·69-s − 2·71-s − 11·75-s + 81-s − 21·89-s + 17·93-s − 14·97-s + 99-s − 19·103-s + ⋯
L(s)  = 1  − 0.577·3-s + 2.23·5-s + 1/3·9-s + 0.301·11-s − 1.29·15-s − 1.45·23-s + 11/5·25-s − 0.192·27-s − 3.05·31-s − 0.174·33-s + 2.95·37-s + 0.745·45-s + 0.875·47-s + 4/7·49-s − 0.961·53-s + 0.674·55-s − 0.911·59-s − 0.610·67-s + 0.842·69-s − 0.237·71-s − 1.27·75-s + 1/9·81-s − 2.22·89-s + 1.76·93-s − 1.42·97-s + 0.100·99-s − 1.87·103-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$C_1$ \( 1 + T \)
11$C_2$ \( 1 - T + p T^{2} \)
good5$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.5.af_o
7$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.7.a_ae
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2^2$ \( 1 - 12 T^{2} + p^{2} T^{4} \) 2.17.a_am
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.19.a_ac
23$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.23.h_bm
29$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \) 2.29.a_bo
31$C_2$$\times$$C_2$ \( ( 1 + 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.31.r_fc
37$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.37.as_fq
41$C_2^2$ \( 1 - 44 T^{2} + p^{2} T^{4} \) 2.41.a_abs
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.43.a_aby
47$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.47.ag_dq
53$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.53.h_em
59$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.59.h_cw
61$C_2^2$ \( 1 + 67 T^{2} + p^{2} T^{4} \) 2.61.a_cp
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.67.f_abq
71$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.71.c_ck
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.73.a_aeg
79$C_2^2$ \( 1 - 116 T^{2} + p^{2} T^{4} \) 2.79.a_aem
83$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \) 2.83.a_cc
89$C_2$$\times$$C_2$ \( ( 1 + 9 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.89.v_la
97$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.97.o_ja
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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

−7.05617647977796659974450820931, −6.42913288956956750108140793276, −6.10960215934441856489982761472, −5.87273928697510775814044871437, −5.48895163075429739243233990600, −5.40068679086499136187984694862, −4.56596697932488471156112357122, −4.20565302313549590914009620850, −3.84185932921907394816988418233, −3.01645482880516455783775028976, −2.57489276769341500282095463428, −1.94577673226723705921931870579, −1.73763824321542830535419672143, −1.10478720416055171153801460686, 0, 1.10478720416055171153801460686, 1.73763824321542830535419672143, 1.94577673226723705921931870579, 2.57489276769341500282095463428, 3.01645482880516455783775028976, 3.84185932921907394816988418233, 4.20565302313549590914009620850, 4.56596697932488471156112357122, 5.40068679086499136187984694862, 5.48895163075429739243233990600, 5.87273928697510775814044871437, 6.10960215934441856489982761472, 6.42913288956956750108140793276, 7.05617647977796659974450820931

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