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Properties

Label 4-2376e2-1.1-c1e2-0-1
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  − 5-s + 2·7-s + 11-s − 12·17-s − 4·19-s + 5·25-s − 3·31-s − 2·35-s − 6·37-s − 8·41-s − 4·43-s − 3·47-s + 7·49-s + 6·53-s − 55-s − 59-s − 4·61-s − 9·67-s − 10·71-s + 4·73-s + 2·77-s − 14·79-s − 12·83-s + 12·85-s + 12·89-s + 4·95-s − 5·97-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.755·7-s + 0.301·11-s − 2.91·17-s − 0.917·19-s + 25-s − 0.538·31-s − 0.338·35-s − 0.986·37-s − 1.24·41-s − 0.609·43-s − 0.437·47-s + 49-s + 0.824·53-s − 0.134·55-s − 0.130·59-s − 0.512·61-s − 1.09·67-s − 1.18·71-s + 0.468·73-s + 0.227·77-s − 1.57·79-s − 1.31·83-s + 1.30·85-s + 1.27·89-s + 0.410·95-s − 0.507·97-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

\(L(1)\) \(\approx\) \(0.5228886392\)
\(L(\frac12)\) \(\approx\) \(0.5228886392\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
good5$C_2^2$ \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) 2.5.b_ae
7$C_2^2$ \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.7.ac_ad
13$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.13.a_an
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.17.m_cs
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.19.e_bq
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.23.a_ax
29$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.29.a_abd
31$C_2^2$ \( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.31.d_aw
37$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.37.g_df
41$C_2^2$ \( 1 + 8 T + 23 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.41.i_x
43$C_2^2$ \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.43.e_abb
47$C_2^2$ \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.47.d_abm
53$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.53.ag_el
59$C_2^2$ \( 1 + T - 58 T^{2} + p T^{3} + p^{2} T^{4} \) 2.59.b_acg
61$C_2^2$ \( 1 + 4 T - 45 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.61.e_abt
67$C_2^2$ \( 1 + 9 T + 14 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.67.j_o
71$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.71.k_gl
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.73.ae_fu
79$C_2^2$ \( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.79.o_en
83$C_2^2$ \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.83.m_cj
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_ig
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) 2.97.f_acu
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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

−8.965279808555825332282854072493, −8.798906510985590295348051129368, −8.623894893299603860733082951216, −7.965940865763841155836631320120, −7.76918567024321494385821978037, −7.04763973269494185689456015082, −6.93499645097866905818441918643, −6.43808816262399078529362581927, −6.35731366261943667076236474491, −5.39917485473641067900512663313, −5.30327016217547919344334888464, −4.59862931936483514733498614241, −4.45467529660208547487658201432, −4.00226461781546426329298417849, −3.60975881244750630995813981129, −2.71241894671690863884798126397, −2.57351874732963785740017992344, −1.66549890763598427469727360096, −1.57029215602321723926386397384, −0.23459990409626774261379472705, 0.23459990409626774261379472705, 1.57029215602321723926386397384, 1.66549890763598427469727360096, 2.57351874732963785740017992344, 2.71241894671690863884798126397, 3.60975881244750630995813981129, 4.00226461781546426329298417849, 4.45467529660208547487658201432, 4.59862931936483514733498614241, 5.30327016217547919344334888464, 5.39917485473641067900512663313, 6.35731366261943667076236474491, 6.43808816262399078529362581927, 6.93499645097866905818441918643, 7.04763973269494185689456015082, 7.76918567024321494385821978037, 7.965940865763841155836631320120, 8.623894893299603860733082951216, 8.798906510985590295348051129368, 8.965279808555825332282854072493

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