Properties

Label 4-1975e2-1.1-c0e2-0-1
Degree $4$
Conductor $3900625$
Sign $1$
Analytic cond. $0.971512$
Root an. cond. $0.992800$
Motivic weight $0$
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  + 2-s + 2·9-s − 11-s + 13-s + 2·18-s − 19-s − 22-s + 23-s + 26-s − 31-s − 32-s − 38-s + 46-s + 2·49-s − 62-s − 64-s + 67-s + 73-s + 2·79-s + 3·81-s − 4·83-s − 89-s + 97-s + 2·98-s − 2·99-s − 101-s + 2·117-s + ⋯
L(s)  = 1  + 2-s + 2·9-s − 11-s + 13-s + 2·18-s − 19-s − 22-s + 23-s + 26-s − 31-s − 32-s − 38-s + 46-s + 2·49-s − 62-s − 64-s + 67-s + 73-s + 2·79-s + 3·81-s − 4·83-s − 89-s + 97-s + 2·98-s − 2·99-s − 101-s + 2·117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3900625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3900625\)    =    \(5^{4} \cdot 79^{2}\)
Sign: $1$
Analytic conductor: \(0.971512\)
Root analytic conductor: \(0.992800\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3900625,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.212601658\)
\(L(\frac12)\) \(\approx\) \(2.212601658\)
\(L(1)\) 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)$
bad5 \( 1 \)
79$C_1$ \( ( 1 - T )^{2} \)
good2$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
3$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
11$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
13$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
19$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
23$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
67$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
83$C_1$ \( ( 1 + T )^{4} \)
89$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
97$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
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

−9.451284998694393232359470644793, −9.270511619461808558546992796173, −8.749909487201758827769763861166, −8.398721715430354508987961026901, −7.909319299788355742157170532638, −7.51867421900615507262213549865, −7.09210009019073384321918082731, −6.84632698003860072509105193584, −6.46906154828270222925825075247, −5.78634651966024102463650429748, −5.43088637505186378559955317445, −5.13154463243713625867282813084, −4.48274635491993578970383196083, −4.38654243336865782138705188928, −3.73063165699825637346463930729, −3.72446257491871022433055357585, −2.84750919889196819187870037443, −2.22913284343448781901232224562, −1.70031292932922563188428338805, −1.02317678223516307668602072964, 1.02317678223516307668602072964, 1.70031292932922563188428338805, 2.22913284343448781901232224562, 2.84750919889196819187870037443, 3.72446257491871022433055357585, 3.73063165699825637346463930729, 4.38654243336865782138705188928, 4.48274635491993578970383196083, 5.13154463243713625867282813084, 5.43088637505186378559955317445, 5.78634651966024102463650429748, 6.46906154828270222925825075247, 6.84632698003860072509105193584, 7.09210009019073384321918082731, 7.51867421900615507262213549865, 7.909319299788355742157170532638, 8.398721715430354508987961026901, 8.749909487201758827769763861166, 9.270511619461808558546992796173, 9.451284998694393232359470644793

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