Properties

Label 4-96e4-1.1-c1e2-0-7
Degree $4$
Conductor $84934656$
Sign $1$
Analytic cond. $5415.50$
Root an. cond. $8.57846$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4·17-s + 8·25-s + 20·41-s − 14·49-s − 12·73-s + 32·89-s + 16·97-s + 32·113-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 24·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 0.970·17-s + 8/5·25-s + 3.12·41-s − 2·49-s − 1.40·73-s + 3.39·89-s + 1.62·97-s + 3.01·113-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.84·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84934656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84934656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(84934656\)    =    \(2^{20} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(5415.50\)
Root analytic conductor: \(8.57846\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{9216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 84934656,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.337846143\)
\(L(\frac12)\) \(\approx\) \(3.337846143\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
good5$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 24 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2^2$ \( 1 - 40 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 24 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2^2$ \( 1 + 56 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2^2$ \( 1 - 120 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.923727233915448102037761211358, −7.56031886274534269277667487008, −7.18987833801168317729950120588, −6.89740607997099250918007257548, −6.38950131364243696130523409317, −6.35782626234089003540580809948, −5.80575485662227328968763569436, −5.62773679579186596038537228936, −4.99114110107998504556230617444, −4.66042170631238922403359692206, −4.55758607511149627204129722926, −4.13073129055239582126466029523, −3.58527982101072897910381238299, −3.17546337267667238586150351273, −2.91346102437549030251606760283, −2.37430739514138678165744667449, −2.02156285873420710333849015251, −1.54168677483258900105429437364, −0.797440792247143384655368984065, −0.54535347116887429231205766424, 0.54535347116887429231205766424, 0.797440792247143384655368984065, 1.54168677483258900105429437364, 2.02156285873420710333849015251, 2.37430739514138678165744667449, 2.91346102437549030251606760283, 3.17546337267667238586150351273, 3.58527982101072897910381238299, 4.13073129055239582126466029523, 4.55758607511149627204129722926, 4.66042170631238922403359692206, 4.99114110107998504556230617444, 5.62773679579186596038537228936, 5.80575485662227328968763569436, 6.35782626234089003540580809948, 6.38950131364243696130523409317, 6.89740607997099250918007257548, 7.18987833801168317729950120588, 7.56031886274534269277667487008, 7.923727233915448102037761211358

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