Properties

Label 4-1134e2-1.1-c1e2-0-39
Degree $4$
Conductor $1285956$
Sign $1$
Analytic cond. $81.9936$
Root an. cond. $3.00915$
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  + 2·2-s + 3·4-s − 3·5-s + 5·7-s + 4·8-s − 6·10-s + 3·11-s − 2·13-s + 10·14-s + 5·16-s − 6·17-s − 2·19-s − 9·20-s + 6·22-s + 6·23-s + 5·25-s − 4·26-s + 15·28-s − 9·29-s − 14·31-s + 6·32-s − 12·34-s − 15·35-s + 10·37-s − 4·38-s − 12·40-s + 4·43-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 1.34·5-s + 1.88·7-s + 1.41·8-s − 1.89·10-s + 0.904·11-s − 0.554·13-s + 2.67·14-s + 5/4·16-s − 1.45·17-s − 0.458·19-s − 2.01·20-s + 1.27·22-s + 1.25·23-s + 25-s − 0.784·26-s + 2.83·28-s − 1.67·29-s − 2.51·31-s + 1.06·32-s − 2.05·34-s − 2.53·35-s + 1.64·37-s − 0.648·38-s − 1.89·40-s + 0.609·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1285956\)    =    \(2^{2} \cdot 3^{8} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(81.9936\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1285956,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.312471756\)
\(L(\frac12)\) \(\approx\) \(5.312471756\)
\(L(\frac{3}{2})\) 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)$
bad2$C_1$ \( ( 1 - T )^{2} \)
3 \( 1 \)
7$C_2$ \( 1 - 5 T + p T^{2} \)
good5$C_2^2$ \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \)
41$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 13 T + 72 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
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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

−10.48540264147949102059502109763, −9.283595896992974442075476352530, −9.093063828339997281485906691852, −8.929852928436489647698857220375, −8.225506060863189531468402953674, −7.70767436708935419406150857337, −7.38724254031940482023510481459, −7.26367851897535541245973555076, −6.82444816796319739797445701865, −5.98214044708989093621057103376, −5.74691084702890531355335921589, −5.15859272159296445261864867054, −4.69444558017019251894814330617, −4.36619023701247332718681478685, −3.95060500408206712452364208494, −3.76474517141537055112774050366, −2.86528573219665802794138618280, −2.16680933705359737953008220227, −1.84675008184175019533426816968, −0.818474004284224395541938613026, 0.818474004284224395541938613026, 1.84675008184175019533426816968, 2.16680933705359737953008220227, 2.86528573219665802794138618280, 3.76474517141537055112774050366, 3.95060500408206712452364208494, 4.36619023701247332718681478685, 4.69444558017019251894814330617, 5.15859272159296445261864867054, 5.74691084702890531355335921589, 5.98214044708989093621057103376, 6.82444816796319739797445701865, 7.26367851897535541245973555076, 7.38724254031940482023510481459, 7.70767436708935419406150857337, 8.225506060863189531468402953674, 8.929852928436489647698857220375, 9.093063828339997281485906691852, 9.283595896992974442075476352530, 10.48540264147949102059502109763

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