Properties

Label 4-80e4-1.1-c1e2-0-23
Degree $4$
Conductor $40960000$
Sign $1$
Analytic cond. $2611.64$
Root an. cond. $7.14872$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 3·9-s − 6·11-s + 6·17-s − 2·19-s − 10·27-s + 12·33-s − 6·41-s − 20·43-s − 14·49-s − 12·51-s + 4·57-s + 12·59-s + 14·67-s + 2·73-s + 20·81-s − 18·83-s + 18·89-s + 20·97-s − 18·99-s − 6·107-s − 18·113-s + 11·121-s + 12·123-s + 127-s + 40·129-s + 131-s + ⋯
L(s)  = 1  − 1.15·3-s + 9-s − 1.80·11-s + 1.45·17-s − 0.458·19-s − 1.92·27-s + 2.08·33-s − 0.937·41-s − 3.04·43-s − 2·49-s − 1.68·51-s + 0.529·57-s + 1.56·59-s + 1.71·67-s + 0.234·73-s + 20/9·81-s − 1.97·83-s + 1.90·89-s + 2.03·97-s − 1.80·99-s − 0.580·107-s − 1.69·113-s + 121-s + 1.08·123-s + 0.0887·127-s + 3.52·129-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(40960000\)    =    \(2^{16} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(2611.64\)
Root analytic conductor: \(7.14872\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{6400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 40960000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good3$C_2^2$ \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2^2$ \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 14 T + 129 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
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$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 18 T + 241 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 18 T + 235 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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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

−7.939724002771569385501129680407, −7.52618259086168918070326595856, −7.02913034123628046649906047335, −6.84095417691906478478812419427, −6.30728773965330542984906644477, −6.17562070056370089250699011503, −5.51439306110800516733455869480, −5.32709264408628589391872086183, −5.16239684480750661246223332754, −4.81256220073585316426354063260, −4.36503149139871097234030572234, −3.73354091258120423012028578223, −3.30101000828655216554427594440, −3.26941960389057798465229049737, −2.36364980743719547641105914999, −2.08268284399654895804492999815, −1.51981137106982608364168941711, −0.952661276307377357255815268635, 0, 0, 0.952661276307377357255815268635, 1.51981137106982608364168941711, 2.08268284399654895804492999815, 2.36364980743719547641105914999, 3.26941960389057798465229049737, 3.30101000828655216554427594440, 3.73354091258120423012028578223, 4.36503149139871097234030572234, 4.81256220073585316426354063260, 5.16239684480750661246223332754, 5.32709264408628589391872086183, 5.51439306110800516733455869480, 6.17562070056370089250699011503, 6.30728773965330542984906644477, 6.84095417691906478478812419427, 7.02913034123628046649906047335, 7.52618259086168918070326595856, 7.939724002771569385501129680407

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