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Properties

Label 4-1320e2-1.1-c1e2-0-31
Degree $4$
Conductor $1742400$
Sign $1$
Analytic cond. $111.096$
Root an. cond. $3.24657$
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  − 3-s + 3·5-s − 2·9-s + 2·11-s − 3·15-s + 10·23-s + 4·25-s + 5·27-s − 31-s − 2·33-s + 6·37-s − 6·45-s + 4·47-s + 8·49-s − 7·53-s + 6·55-s + 7·67-s − 10·69-s − 71-s − 4·75-s + 81-s + 5·89-s + 93-s − 4·97-s − 4·99-s − 11·103-s − 6·111-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.34·5-s − 2/3·9-s + 0.603·11-s − 0.774·15-s + 2.08·23-s + 4/5·25-s + 0.962·27-s − 0.179·31-s − 0.348·33-s + 0.986·37-s − 0.894·45-s + 0.583·47-s + 8/7·49-s − 0.961·53-s + 0.809·55-s + 0.855·67-s − 1.20·69-s − 0.118·71-s − 0.461·75-s + 1/9·81-s + 0.529·89-s + 0.103·93-s − 0.406·97-s − 0.402·99-s − 1.08·103-s − 0.569·111-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1742400\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(111.096\)
Root analytic conductor: \(3.24657\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1742400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.488072036\)
\(L(\frac12)\) \(\approx\) \(2.488072036\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + p T^{2} \)
5$C_2$ \( 1 - 3 T + p T^{2} \)
11$C_2$ \( 1 - 2 T + p T^{2} \)
good7$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.7.a_ai
13$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \) 2.13.a_ar
17$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.17.a_k
19$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.19.a_e
23$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.23.ak_cs
29$C_2^2$ \( 1 + 36 T^{2} + p^{2} T^{4} \) 2.29.a_bk
31$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.31.b_ce
37$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.ag_c
41$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.41.a_aw
43$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \) 2.43.a_u
47$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.47.ae_r
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.53.h_bc
59$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.59.a_ada
61$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.61.a_az
67$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.67.ah_em
71$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.71.b_cs
73$C_2^2$ \( 1 - 39 T^{2} + p^{2} T^{4} \) 2.73.a_abn
79$C_2^2$ \( 1 + 48 T^{2} + p^{2} T^{4} \) 2.79.a_bw
83$C_2^2$ \( 1 - 35 T^{2} + p^{2} T^{4} \) 2.83.a_abj
89$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.89.af_ca
97$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.97.e_hm
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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

−7.81490956631735883196439282134, −7.24778325597194871990959210795, −6.83181829297048305928022940315, −6.50261056136448259948151625674, −6.07111342419992463624540374903, −5.66782266903990255049563215136, −5.34227182835164058343884335647, −4.90020439549167858036084805778, −4.42851366742501083311110067889, −3.77192042006366392557449347373, −3.08868726901837514029882940909, −2.69784764105751400513264512444, −2.11645839465009846006285442470, −1.33794155319946546676789167247, −0.74130238237165472085202668292, 0.74130238237165472085202668292, 1.33794155319946546676789167247, 2.11645839465009846006285442470, 2.69784764105751400513264512444, 3.08868726901837514029882940909, 3.77192042006366392557449347373, 4.42851366742501083311110067889, 4.90020439549167858036084805778, 5.34227182835164058343884335647, 5.66782266903990255049563215136, 6.07111342419992463624540374903, 6.50261056136448259948151625674, 6.83181829297048305928022940315, 7.24778325597194871990959210795, 7.81490956631735883196439282134

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