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Properties

Label 4-3024e2-1.1-c1e2-0-39
Degree $4$
Conductor $9144576$
Sign $1$
Analytic cond. $583.066$
Root an. cond. $4.91393$
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  − 7-s + 16·19-s + 7·25-s − 12·29-s − 10·31-s + 8·37-s + 12·47-s − 6·49-s − 6·53-s + 24·59-s + 30·83-s + 32·103-s + 40·109-s − 24·113-s − 5·121-s + 127-s + 131-s − 16·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 14·169-s + 173-s + ⋯
L(s)  = 1  − 0.377·7-s + 3.67·19-s + 7/5·25-s − 2.22·29-s − 1.79·31-s + 1.31·37-s + 1.75·47-s − 6/7·49-s − 0.824·53-s + 3.12·59-s + 3.29·83-s + 3.15·103-s + 3.83·109-s − 2.25·113-s − 0.454·121-s + 0.0887·127-s + 0.0873·131-s − 1.38·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.07·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9144576\)    =    \(2^{8} \cdot 3^{6} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(583.066\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9144576,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.141486335\)
\(L(\frac12)\) \(\approx\) \(3.141486335\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 + T + p T^{2} \)
good5$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \) 2.5.a_ah
11$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \) 2.11.a_f
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$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.19.aq_dy
23$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.23.a_c
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.29.m_dq
31$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.31.k_dj
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.37.ai_dm
41$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.41.a_abi
43$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \) 2.43.a_acw
47$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.47.am_fa
53$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.53.g_el
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.59.ay_kc
61$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.61.a_cs
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.67.a_aes
71$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.71.a_afa
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) 2.73.a_afn
79$C_2^2$ \( 1 - 146 T^{2} + p^{2} T^{4} \) 2.79.a_afq
83$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \) 2.83.abe_pb
89$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.89.a_acs
97$C_2^2$ \( 1 - 119 T^{2} + p^{2} T^{4} \) 2.97.a_aep
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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

−8.947360113500873729447056366871, −8.834815641402780658271814583611, −7.916329038464943567327567595441, −7.70619756058522834138633991996, −7.37524741406079458456087945294, −7.25180959101029417179268977189, −6.75254699102914385100327303135, −6.18465256321573065873237736592, −5.80704962570355589096398723037, −5.41297759994517055010377640641, −5.11983744997274675325587775264, −4.89211396848668252674936444772, −4.02076114841670281127736337429, −3.71909456423703670078772308042, −3.23581104950657391667893110928, −3.10092649111417880249869311003, −2.27098717715432984836517366330, −1.87391491904726699300006483400, −0.976999593662215870090168729665, −0.70535290475981124489117592454, 0.70535290475981124489117592454, 0.976999593662215870090168729665, 1.87391491904726699300006483400, 2.27098717715432984836517366330, 3.10092649111417880249869311003, 3.23581104950657391667893110928, 3.71909456423703670078772308042, 4.02076114841670281127736337429, 4.89211396848668252674936444772, 5.11983744997274675325587775264, 5.41297759994517055010377640641, 5.80704962570355589096398723037, 6.18465256321573065873237736592, 6.75254699102914385100327303135, 7.25180959101029417179268977189, 7.37524741406079458456087945294, 7.70619756058522834138633991996, 7.916329038464943567327567595441, 8.834815641402780658271814583611, 8.947360113500873729447056366871

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