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Properties

Label 4-2178e2-1.1-c1e2-0-9
Degree $4$
Conductor $4743684$
Sign $1$
Analytic cond. $302.461$
Root an. cond. $4.17030$
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 + 16-s − 2·20-s − 2·23-s − 6·25-s + 12·31-s + 4·37-s + 22·47-s + 10·49-s + 14·53-s − 64-s + 4·67-s + 2·71-s + 2·80-s + 16·89-s + 2·92-s − 16·97-s + 6·100-s + 12·103-s − 4·115-s − 12·124-s − 22·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 1/2·4-s + 0.894·5-s + 1/4·16-s − 0.447·20-s − 0.417·23-s − 6/5·25-s + 2.15·31-s + 0.657·37-s + 3.20·47-s + 10/7·49-s + 1.92·53-s − 1/8·64-s + 0.488·67-s + 0.237·71-s + 0.223·80-s + 1.69·89-s + 0.208·92-s − 1.62·97-s + 3/5·100-s + 1.18·103-s − 0.373·115-s − 1.07·124-s − 1.96·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4743684 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4743684 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4743684\)    =    \(2^{2} \cdot 3^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(302.461\)
Root analytic conductor: \(4.17030\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4743684,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.760130145\)
\(L(\frac12)\) \(\approx\) \(2.760130145\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 \)
good5$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) 2.5.ac_k
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
13$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.13.a_o
17$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.17.a_g
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$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.29.a_abe
31$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.31.am_de
37$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.37.ae_bq
41$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.41.a_abe
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.43.a_aby
47$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) 2.47.aw_ig
53$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.53.ao_fa
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.a_dy
61$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.61.a_bu
67$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.67.ae_fe
71$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.71.ac_dq
73$C_2^2$ \( 1 + 42 T^{2} + p^{2} T^{4} \) 2.73.a_bq
79$C_2^2$ \( 1 + 118 T^{2} + p^{2} T^{4} \) 2.79.a_eo
83$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.83.a_s
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.89.aq_je
97$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.q_io
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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

−7.36678845502632342168672112691, −6.97027963609281113992857566403, −6.37160089892364298835185624337, −6.06117599566428992953359389779, −5.70566913052718427955866520455, −5.44000211394057449132972881597, −4.87996718478394742321924105992, −4.31009317745299156776839117067, −4.03858693828048346840417455721, −3.66088401480996124109882840051, −2.81165514396817769247549221177, −2.38370510004962086118082422263, −2.11136272948185817518093069036, −1.13597066474028100610125612365, −0.66853672832944652911592851410, 0.66853672832944652911592851410, 1.13597066474028100610125612365, 2.11136272948185817518093069036, 2.38370510004962086118082422263, 2.81165514396817769247549221177, 3.66088401480996124109882840051, 4.03858693828048346840417455721, 4.31009317745299156776839117067, 4.87996718478394742321924105992, 5.44000211394057449132972881597, 5.70566913052718427955866520455, 6.06117599566428992953359389779, 6.37160089892364298835185624337, 6.97027963609281113992857566403, 7.36678845502632342168672112691

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