Properties

Label 4-5e4-1.1-c7e2-0-4
Degree $4$
Conductor $625$
Sign $1$
Analytic cond. $60.9902$
Root an. cond. $2.79457$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 140·4-s − 3.33e3·9-s − 1.36e4·11-s + 3.21e3·16-s − 1.37e4·19-s − 5.11e4·29-s + 1.64e5·31-s + 4.66e5·36-s − 1.06e6·41-s + 1.91e6·44-s − 1.47e6·49-s − 2.87e6·59-s + 2.76e6·61-s + 1.84e6·64-s − 9.63e5·71-s + 1.92e6·76-s + 2.11e6·79-s + 6.30e6·81-s − 1.12e7·89-s + 4.54e7·99-s + 1.02e7·101-s + 4.02e7·109-s + 7.16e6·116-s + 1.00e8·121-s − 2.29e7·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 1.09·4-s − 1.52·9-s − 3.09·11-s + 0.196·16-s − 0.458·19-s − 0.389·29-s + 0.990·31-s + 1.66·36-s − 2.41·41-s + 3.38·44-s − 1.78·49-s − 1.82·59-s + 1.55·61-s + 0.879·64-s − 0.319·71-s + 0.501·76-s + 0.483·79-s + 1.31·81-s − 1.69·89-s + 4.71·99-s + 0.993·101-s + 2.97·109-s + 0.426·116-s + 5.17·121-s − 1.08·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(625\)    =    \(5^{4}\)
Sign: $1$
Analytic conductor: \(60.9902\)
Root analytic conductor: \(2.79457\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 625,\ (\ :7/2, 7/2),\ 1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) 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)$
bad5 \( 1 \)
good2$C_2^2$ \( 1 + 35 p^{2} T^{2} + p^{14} T^{4} \)
3$C_2^2$ \( 1 + 370 p^{2} T^{2} + p^{14} T^{4} \)
7$C_2^2$ \( 1 + 1470650 T^{2} + p^{14} T^{4} \)
11$C_2$ \( ( 1 + 6828 T + p^{7} T^{2} )^{2} \)
13$C_2^2$ \( 1 + 22562810 T^{2} + p^{14} T^{4} \)
17$C_2^2$ \( 1 + 574764770 T^{2} + p^{14} T^{4} \)
19$C_2$ \( ( 1 + 6860 T + p^{7} T^{2} )^{2} \)
23$C_2^2$ \( 1 + 5955848090 T^{2} + p^{14} T^{4} \)
29$C_2$ \( ( 1 + 25590 T + p^{7} T^{2} )^{2} \)
31$C_2$ \( ( 1 - 82112 T + p^{7} T^{2} )^{2} \)
37$C_2^2$ \( 1 + 139899246410 T^{2} + p^{14} T^{4} \)
41$C_2$ \( ( 1 + 533118 T + p^{7} T^{2} )^{2} \)
43$C_2^2$ \( 1 + 41047812050 T^{2} + p^{14} T^{4} \)
47$C_2^2$ \( 1 + 1013212289930 T^{2} + p^{14} T^{4} \)
53$C_2^2$ \( 1 + 2002060594730 T^{2} + p^{14} T^{4} \)
59$C_2$ \( ( 1 + 1438980 T + p^{7} T^{2} )^{2} \)
61$C_2$ \( ( 1 - 1381022 T + p^{7} T^{2} )^{2} \)
67$C_2^2$ \( 1 + 4750924642370 T^{2} + p^{14} T^{4} \)
71$C_2$ \( ( 1 + 481608 T + p^{7} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 19886077213490 T^{2} + p^{14} T^{4} \)
79$C_2$ \( ( 1 - 1059760 T + p^{7} T^{2} )^{2} \)
83$C_2^2$ \( 1 + 47492314121570 T^{2} + p^{14} T^{4} \)
89$C_2$ \( ( 1 + 5644170 T + p^{7} T^{2} )^{2} \)
97$C_2^2$ \( 1 + 17378330046530 T^{2} + p^{14} T^{4} \)
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

−15.40354952650929016516590354035, −15.33232123919012325496933600721, −14.16598149052077995774652291165, −13.84481076462811257914444539010, −12.99112158642373720012132355451, −12.94567428917971928457347779358, −11.74114402062395342599699503693, −11.03241876730854626360171447453, −10.32251569245436240376323332977, −9.748932273899808354531736513173, −8.487378557301427057655142049864, −8.434667252026757798950217634537, −7.58046908619093817137203735020, −6.17079253694754335447791025510, −5.15746859765721955026539699123, −4.91916063312236110405298149234, −3.25772023830105731930052251909, −2.41473398784758606798274498933, 0, 0, 2.41473398784758606798274498933, 3.25772023830105731930052251909, 4.91916063312236110405298149234, 5.15746859765721955026539699123, 6.17079253694754335447791025510, 7.58046908619093817137203735020, 8.434667252026757798950217634537, 8.487378557301427057655142049864, 9.748932273899808354531736513173, 10.32251569245436240376323332977, 11.03241876730854626360171447453, 11.74114402062395342599699503693, 12.94567428917971928457347779358, 12.99112158642373720012132355451, 13.84481076462811257914444539010, 14.16598149052077995774652291165, 15.33232123919012325496933600721, 15.40354952650929016516590354035

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