Properties

Label 4-48e4-1.1-c2e2-0-14
Degree $4$
Conductor $5308416$
Sign $1$
Analytic cond. $3941.25$
Root an. cond. $7.92334$
Motivic weight $2$
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  − 32·11-s + 48·17-s − 48·19-s + 34·25-s − 16·41-s + 112·43-s + 82·49-s + 64·59-s − 160·67-s + 132·73-s − 32·83-s + 288·89-s + 188·97-s + 192·107-s − 160·113-s + 526·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 334·169-s + 173-s + ⋯
L(s)  = 1  − 2.90·11-s + 2.82·17-s − 2.52·19-s + 1.35·25-s − 0.390·41-s + 2.60·43-s + 1.67·49-s + 1.08·59-s − 2.38·67-s + 1.80·73-s − 0.385·83-s + 3.23·89-s + 1.93·97-s + 1.79·107-s − 1.41·113-s + 4.34·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 1.97·169-s + 0.00578·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5308416\)    =    \(2^{16} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(3941.25\)
Root analytic conductor: \(7.92334\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5308416,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.568027168\)
\(L(\frac12)\) \(\approx\) \(2.568027168\)
\(L(2)\) 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 \( 1 \)
3 \( 1 \)
good5$C_2^2$ \( 1 - 34 T^{2} + p^{4} T^{4} \)
7$C_2^2$ \( 1 - 82 T^{2} + p^{4} T^{4} \)
11$C_2$ \( ( 1 + 16 T + p^{2} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 334 T^{2} + p^{4} T^{4} \)
17$C_2$ \( ( 1 - 24 T + p^{2} T^{2} )^{2} \)
19$C_2$ \( ( 1 + 24 T + p^{2} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 34 T^{2} + p^{4} T^{4} \)
29$C_2^2$ \( 1 + 254 T^{2} + p^{4} T^{4} \)
31$C_2^2$ \( 1 + 782 T^{2} + p^{4} T^{4} \)
37$C_2^2$ \( 1 - 2414 T^{2} + p^{4} T^{4} \)
41$C_2$ \( ( 1 + 8 T + p^{2} T^{2} )^{2} \)
43$C_2$ \( ( 1 - 56 T + p^{2} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 3394 T^{2} + p^{4} T^{4} \)
53$C_2^2$ \( 1 - 4322 T^{2} + p^{4} T^{4} \)
59$C_2$ \( ( 1 - 32 T + p^{2} T^{2} )^{2} \)
61$C_2^2$ \( 1 - 3598 T^{2} + p^{4} T^{4} \)
67$C_2$ \( ( 1 + 80 T + p^{2} T^{2} )^{2} \)
71$C_2^2$ \( 1 + 6302 T^{2} + p^{4} T^{4} \)
73$C_2$ \( ( 1 - 66 T + p^{2} T^{2} )^{2} \)
79$C_2^2$ \( 1 - 12082 T^{2} + p^{4} T^{4} \)
83$C_2$ \( ( 1 + 16 T + p^{2} T^{2} )^{2} \)
89$C_2$ \( ( 1 - 144 T + p^{2} T^{2} )^{2} \)
97$C_2$ \( ( 1 - 94 T + p^{2} T^{2} )^{2} \)
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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

−8.816298286089262062554985256513, −8.767789754573431247715339430530, −8.067592988002811695964943160982, −7.83856922725299487132560381741, −7.54790542998400610116573078718, −7.40226390920708554778478015529, −6.65345211061052491947082776848, −6.17072651892034610064154118092, −5.87388618258039897259813189933, −5.43072525782643909796274546877, −5.00326315968549185548522688770, −4.91628536197234334768379733260, −4.06106532895167740169035536969, −3.83495284578629499321075635255, −2.98189376801723804257405299760, −2.82995013549442029424772120197, −2.34295549478418174830223208571, −1.83113177141942895798895767702, −0.78170241530301435121595572573, −0.53419145049410065521816001180, 0.53419145049410065521816001180, 0.78170241530301435121595572573, 1.83113177141942895798895767702, 2.34295549478418174830223208571, 2.82995013549442029424772120197, 2.98189376801723804257405299760, 3.83495284578629499321075635255, 4.06106532895167740169035536969, 4.91628536197234334768379733260, 5.00326315968549185548522688770, 5.43072525782643909796274546877, 5.87388618258039897259813189933, 6.17072651892034610064154118092, 6.65345211061052491947082776848, 7.40226390920708554778478015529, 7.54790542998400610116573078718, 7.83856922725299487132560381741, 8.067592988002811695964943160982, 8.767789754573431247715339430530, 8.816298286089262062554985256513

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