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Properties

Label 4-1776e2-1.1-c1e2-0-17
Degree $4$
Conductor $3154176$
Sign $1$
Analytic cond. $201.112$
Root an. cond. $3.76582$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·5-s − 3·7-s − 4·11-s + 3·13-s + 2·15-s − 4·19-s + 3·21-s − 12·23-s + 5·25-s + 27-s − 2·31-s + 4·33-s + 6·35-s − 10·37-s − 3·39-s + 10·43-s + 8·47-s + 7·49-s − 6·53-s + 8·55-s + 4·57-s − 10·59-s + 2·61-s − 6·65-s + 3·67-s + 12·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s − 1.13·7-s − 1.20·11-s + 0.832·13-s + 0.516·15-s − 0.917·19-s + 0.654·21-s − 2.50·23-s + 25-s + 0.192·27-s − 0.359·31-s + 0.696·33-s + 1.01·35-s − 1.64·37-s − 0.480·39-s + 1.52·43-s + 1.16·47-s + 49-s − 0.824·53-s + 1.07·55-s + 0.529·57-s − 1.30·59-s + 0.256·61-s − 0.744·65-s + 0.366·67-s + 1.44·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3154176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3154176 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3154176\)    =    \(2^{8} \cdot 3^{2} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(201.112\)
Root analytic conductor: \(3.76582\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 3154176,\ (\ :1/2, 1/2),\ 1)\)

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$C_2$ \( 1 + T + T^{2} \)
37$C_2$ \( 1 + 10 T + p T^{2} \)
good5$C_2^2$ \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.5.c_ab
7$C_2^2$ \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.7.d_c
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.11.e_ba
13$C_2^2$ \( 1 - 3 T - 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.13.ad_ae
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.17.a_ar
19$C_2^2$ \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.19.e_ad
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.23.m_de
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.31.c_cl
41$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.41.a_abp
43$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.43.ak_eh
47$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.47.ai_eg
53$C_2^2$ \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.53.g_ar
59$C_2^2$ \( 1 + 10 T + 41 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.59.k_bp
61$C_2^2$ \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.61.ac_acf
67$C_2^2$ \( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.67.ad_acg
71$C_2^2$ \( 1 - 12 T + 73 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.71.am_cv
73$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.73.ac_fr
79$C_2^2$ \( 1 - 15 T + 146 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.79.ap_fq
83$C_2^2$ \( 1 + 4 T - 67 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.83.e_acp
89$C_2^2$ \( 1 - 10 T + 11 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.89.ak_l
97$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.97.as_kp
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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

−9.144958641779575245471428456871, −8.608759659987445664342600276099, −8.241704870916698251166373185686, −7.933673284679220373098347291878, −7.59296690001089490838927234669, −7.09898113270421908889601859681, −6.47209193305976940243703729104, −6.44976214089005832354443666975, −5.88383245311739629423630491750, −5.49221749279152383590231680782, −5.11198429733476077929360547703, −4.46063008860799477537917452553, −3.87308562929304745378773273624, −3.82975793491517599382679014625, −3.20694390772441205078424911580, −2.49897486726568506014194927938, −2.16771212218297486003073459570, −1.13753888445410453289471564460, 0, 0, 1.13753888445410453289471564460, 2.16771212218297486003073459570, 2.49897486726568506014194927938, 3.20694390772441205078424911580, 3.82975793491517599382679014625, 3.87308562929304745378773273624, 4.46063008860799477537917452553, 5.11198429733476077929360547703, 5.49221749279152383590231680782, 5.88383245311739629423630491750, 6.44976214089005832354443666975, 6.47209193305976940243703729104, 7.09898113270421908889601859681, 7.59296690001089490838927234669, 7.933673284679220373098347291878, 8.241704870916698251166373185686, 8.608759659987445664342600276099, 9.144958641779575245471428456871

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