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Properties

Label 4-2442e2-1.1-c1e2-0-5
Degree $4$
Conductor $5963364$
Sign $1$
Analytic cond. $380.229$
Root an. cond. $4.41582$
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  − 4-s − 2·5-s + 9-s + 2·11-s + 16-s + 2·20-s − 2·23-s + 2·25-s + 8·31-s − 36-s − 6·37-s − 2·44-s − 2·45-s + 8·47-s − 2·49-s + 4·53-s − 4·55-s + 18·59-s − 64-s + 4·67-s − 2·80-s + 81-s + 14·89-s + 2·92-s + 2·99-s − 2·100-s − 16·103-s + ⋯
L(s)  = 1  − 1/2·4-s − 0.894·5-s + 1/3·9-s + 0.603·11-s + 1/4·16-s + 0.447·20-s − 0.417·23-s + 2/5·25-s + 1.43·31-s − 1/6·36-s − 0.986·37-s − 0.301·44-s − 0.298·45-s + 1.16·47-s − 2/7·49-s + 0.549·53-s − 0.539·55-s + 2.34·59-s − 1/8·64-s + 0.488·67-s − 0.223·80-s + 1/9·81-s + 1.48·89-s + 0.208·92-s + 0.201·99-s − 1/5·100-s − 1.57·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5963364 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5963364 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5963364\)    =    \(2^{2} \cdot 3^{2} \cdot 11^{2} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(380.229\)
Root analytic conductor: \(4.41582\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5963364,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.719538192\)
\(L(\frac12)\) \(\approx\) \(1.719538192\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
11$C_2$ \( 1 - 2 T + p T^{2} \)
37$C_2$ \( 1 + 6 T + p T^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.c_c
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
13$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.13.a_ao
17$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.17.a_aw
19$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.19.a_g
23$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.c_bm
29$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \) 2.29.a_abm
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) 2.31.ai_ck
41$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \) 2.41.a_abm
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.43.a_aby
47$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.47.ai_bu
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.53.ae_dq
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.59.as_hi
61$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.61.a_ec
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.ae_dy
71$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.71.a_ec
73$C_2^2$ \( 1 + 106 T^{2} + p^{2} T^{4} \) 2.73.a_ec
79$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \) 2.79.a_acw
83$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.83.a_bi
89$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.89.ao_fq
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.97.a_hi
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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

−7.13177316140564855614140385449, −6.85386186979449429857342839229, −6.61584133798298695423464441160, −6.03491195581629744691228448953, −5.49582489298661908936756052615, −5.25255484691773420887169977281, −4.65605117790123903931376850521, −4.22785710959710160762186262862, −3.99086438363092641803356193680, −3.57703391521000060641800902114, −3.04019693838766176747399595911, −2.45187002292227506006957788256, −1.84911971324680938644948352520, −1.06383159783776234448266791228, −0.53248373691956833641989142458, 0.53248373691956833641989142458, 1.06383159783776234448266791228, 1.84911971324680938644948352520, 2.45187002292227506006957788256, 3.04019693838766176747399595911, 3.57703391521000060641800902114, 3.99086438363092641803356193680, 4.22785710959710160762186262862, 4.65605117790123903931376850521, 5.25255484691773420887169977281, 5.49582489298661908936756052615, 6.03491195581629744691228448953, 6.61584133798298695423464441160, 6.85386186979449429857342839229, 7.13177316140564855614140385449

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