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Properties

Label 4-364e2-1.1-c1e2-0-5
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s + 4·7-s + 3·9-s + 11-s + 2·13-s + 15-s + 5·17-s + 7·19-s − 4·21-s + 23-s + 5·25-s − 8·27-s + 20·29-s − 3·31-s − 33-s − 4·35-s + 11·37-s − 2·39-s − 20·41-s − 16·43-s − 3·45-s − 9·47-s + 9·49-s − 5·51-s + 9·53-s − 55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1.51·7-s + 9-s + 0.301·11-s + 0.554·13-s + 0.258·15-s + 1.21·17-s + 1.60·19-s − 0.872·21-s + 0.208·23-s + 25-s − 1.53·27-s + 3.71·29-s − 0.538·31-s − 0.174·33-s − 0.676·35-s + 1.80·37-s − 0.320·39-s − 3.12·41-s − 2.43·43-s − 0.447·45-s − 1.31·47-s + 9/7·49-s − 0.700·51-s + 1.23·53-s − 0.134·55-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

\(L(1)\) \(\approx\) \(1.840213715\)
\(L(\frac12)\) \(\approx\) \(1.840213715\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) 2.5.b_ae
11$C_2^2$ \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \) 2.11.ab_ak
17$C_2^2$ \( 1 - 5 T + 8 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.17.af_i
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.19.ah_be
23$C_2^2$ \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \) 2.23.ab_aw
29$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.29.au_gc
31$C_2^2$ \( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.31.d_aw
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.37.al_dg
41$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.41.u_ha
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.43.q_fu
47$C_2^2$ \( 1 + 9 T + 34 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.47.j_bi
53$C_2^2$ \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.53.aj_bc
59$C_2^2$ \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.59.d_aby
61$C_2^2$ \( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.61.l_ci
67$C_2^2$ \( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.67.h_as
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.71.q_hy
73$C_2^2$ \( 1 - 3 T - 64 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.73.ad_acm
79$C_2^2$ \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) 2.79.ab_ada
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.83.i_ha
89$C_2^2$ \( 1 + 9 T - 8 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.89.j_ai
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

−11.58691479531909708624145173386, −11.53510397136054144277286312365, −10.68540691769198290440766634799, −10.29228825986606117490130983387, −10.01140677682857241135101230196, −9.509753793556838884887677339583, −8.574618356884655163591066707820, −8.465221014482521346219318506177, −7.87014671581200012252903570335, −7.53762956378987279161791299314, −6.71354762204812753123946089907, −6.67683100681408548193944613037, −5.67700845498465212838917227842, −5.22239456930474475640582670280, −4.65854418158152324935410839958, −4.46402841445310318677942759128, −3.39075986614740874725387306261, −3.01531097051232033301670028275, −1.38071571072859150145079089404, −1.32259090172642132570892459143, 1.32259090172642132570892459143, 1.38071571072859150145079089404, 3.01531097051232033301670028275, 3.39075986614740874725387306261, 4.46402841445310318677942759128, 4.65854418158152324935410839958, 5.22239456930474475640582670280, 5.67700845498465212838917227842, 6.67683100681408548193944613037, 6.71354762204812753123946089907, 7.53762956378987279161791299314, 7.87014671581200012252903570335, 8.465221014482521346219318506177, 8.574618356884655163591066707820, 9.509753793556838884887677339583, 10.01140677682857241135101230196, 10.29228825986606117490130983387, 10.68540691769198290440766634799, 11.53510397136054144277286312365, 11.58691479531909708624145173386

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