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Properties

Label 4-364e2-1.1-c1e2-0-1
Degree $4$
Conductor $132496$
Sign $1$
Analytic cond. $8.44805$
Root an. cond. $1.70486$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 3·5-s − 4·7-s + 3·9-s − 3·11-s + 2·13-s − 3·15-s − 3·17-s − 5·19-s + 4·21-s + 9·23-s + 5·25-s − 8·27-s − 12·29-s + 31-s + 3·33-s − 12·35-s + 7·37-s − 2·39-s + 12·41-s + 16·43-s + 9·45-s + 3·47-s + 9·49-s + 3·51-s + 9·53-s − 9·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.34·5-s − 1.51·7-s + 9-s − 0.904·11-s + 0.554·13-s − 0.774·15-s − 0.727·17-s − 1.14·19-s + 0.872·21-s + 1.87·23-s + 25-s − 1.53·27-s − 2.22·29-s + 0.179·31-s + 0.522·33-s − 2.02·35-s + 1.15·37-s − 0.320·39-s + 1.87·41-s + 2.43·43-s + 1.34·45-s + 0.437·47-s + 9/7·49-s + 0.420·51-s + 1.23·53-s − 1.21·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 132496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(132496\)    =    \(2^{4} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(8.44805\)
Root analytic conductor: \(1.70486\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 132496,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.323855984\)
\(L(\frac12)\) \(\approx\) \(1.323855984\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
13$C_1$ \( ( 1 - T )^{2} \)
good3$C_2^2$ \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) 2.3.b_ac
5$C_2^2$ \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.5.ad_e
11$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.11.d_ac
17$C_2^2$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.17.d_ai
19$C_2^2$ \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.19.f_g
23$C_2^2$ \( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.23.aj_cg
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.29.m_dq
31$C_2^2$ \( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} \) 2.31.ab_abe
37$C_2^2$ \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.37.ah_m
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.41.am_eo
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.43.aq_fu
47$C_2^2$ \( 1 - 3 T - 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.47.ad_abm
53$C_2^2$ \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.53.aj_bc
59$C_2^2$ \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.59.aj_w
61$C_2^2$ \( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.61.l_ci
67$C_2^2$ \( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.67.an_dy
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.ah_ay
79$C_2^2$ \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) 2.79.ab_ada
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.83.y_ly
89$C_2^2$ \( 1 - 3 T - 80 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.89.ad_adc
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.97.ae_hq
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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

−11.26495528194578218750630804751, −11.11112917093124904531287173862, −10.86674333680401948221816533408, −10.13697613796260715727684269978, −9.918709980197902933394250842318, −9.385886066757878091422732086829, −9.055015225537876847418070447786, −8.732470797631734100857150863621, −7.64915161171656918253164173594, −7.25866500464301775145806264114, −6.92793761986758153809297199879, −6.05090578804677772670637098430, −5.99114482014397103010466644691, −5.62568345168574327299096182741, −4.75347023712293927990321445705, −4.15557552790868017060073521305, −3.53125869113708898095507178894, −2.49920421661630549385620254954, −2.19820704604892823547842425019, −0.822400897861456104223111575487, 0.822400897861456104223111575487, 2.19820704604892823547842425019, 2.49920421661630549385620254954, 3.53125869113708898095507178894, 4.15557552790868017060073521305, 4.75347023712293927990321445705, 5.62568345168574327299096182741, 5.99114482014397103010466644691, 6.05090578804677772670637098430, 6.92793761986758153809297199879, 7.25866500464301775145806264114, 7.64915161171656918253164173594, 8.732470797631734100857150863621, 9.055015225537876847418070447786, 9.385886066757878091422732086829, 9.918709980197902933394250842318, 10.13697613796260715727684269978, 10.86674333680401948221816533408, 11.11112917093124904531287173862, 11.26495528194578218750630804751

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