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Properties

Label 4-1512e2-1.1-c1e2-0-15
Degree $4$
Conductor $2286144$
Sign $1$
Analytic cond. $145.766$
Root an. cond. $3.47467$
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  − 2·4-s + 12·11-s + 4·16-s + 6·17-s + 4·19-s − 25-s − 6·41-s − 2·43-s − 24·44-s + 49-s + 18·59-s − 8·64-s − 8·67-s − 12·68-s + 4·73-s − 8·76-s + 6·83-s + 12·89-s − 20·97-s + 2·100-s + 12·107-s − 24·113-s + 86·121-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 4-s + 3.61·11-s + 16-s + 1.45·17-s + 0.917·19-s − 1/5·25-s − 0.937·41-s − 0.304·43-s − 3.61·44-s + 1/7·49-s + 2.34·59-s − 64-s − 0.977·67-s − 1.45·68-s + 0.468·73-s − 0.917·76-s + 0.658·83-s + 1.27·89-s − 2.03·97-s + 1/5·100-s + 1.16·107-s − 2.25·113-s + 7.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2286144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2286144 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2286144\)    =    \(2^{6} \cdot 3^{6} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(145.766\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2286144,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.776148896\)
\(L(\frac12)\) \(\approx\) \(2.776148896\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
3 \( 1 \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.5.a_b
11$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.11.am_cg
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.17.ag_br
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.19.ae_bq
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.37.a_z
41$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.41.g_dn
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.43.c_dj
47$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.47.a_n
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.59.as_hr
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.a_w
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.67.i_fu
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.73.ae_fu
79$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.79.a_gb
83$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.83.ag_gt
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_ig
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.97.u_li
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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

−7.46920162523798836927398059161, −7.46477788862935127887709071190, −6.70886439785070977996370612854, −6.47854455196725766229052156279, −6.04902466517769458594559190180, −5.50095261947846622681824006136, −5.13832633779419862585354738180, −4.61450873119863775415893038321, −3.91235022223803928204843828241, −3.79566092998735399945677628017, −3.53265306563594616744874259894, −2.80733365210463832197294691089, −1.69414023663856774565794959548, −1.28316529913330947219949011653, −0.815221262012707277849084396531, 0.815221262012707277849084396531, 1.28316529913330947219949011653, 1.69414023663856774565794959548, 2.80733365210463832197294691089, 3.53265306563594616744874259894, 3.79566092998735399945677628017, 3.91235022223803928204843828241, 4.61450873119863775415893038321, 5.13832633779419862585354738180, 5.50095261947846622681824006136, 6.04902466517769458594559190180, 6.47854455196725766229052156279, 6.70886439785070977996370612854, 7.46477788862935127887709071190, 7.46920162523798836927398059161

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