↑

Properties

Label 4-845e2-1.1-c1e2-0-1
Degree $4$
Conductor $714025$
Sign $1$
Analytic cond. $45.5268$
Root an. cond. $2.59756$
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  + 2·3-s − 4-s + 2·5-s − 4·7-s + 6·11-s − 2·12-s + 4·15-s − 3·16-s + 2·19-s − 2·20-s − 8·21-s + 6·23-s + 3·25-s − 2·27-s + 4·28-s − 12·29-s − 10·31-s + 12·33-s − 8·35-s + 8·37-s + 10·43-s − 6·44-s − 12·47-s − 6·48-s − 2·49-s + 12·55-s + 4·57-s + ⋯
L(s)  = 1  + 1.15·3-s − 1/2·4-s + 0.894·5-s − 1.51·7-s + 1.80·11-s − 0.577·12-s + 1.03·15-s − 3/4·16-s + 0.458·19-s − 0.447·20-s − 1.74·21-s + 1.25·23-s + 3/5·25-s − 0.384·27-s + 0.755·28-s − 2.22·29-s − 1.79·31-s + 2.08·33-s − 1.35·35-s + 1.31·37-s + 1.52·43-s − 0.904·44-s − 1.75·47-s − 0.866·48-s − 2/7·49-s + 1.61·55-s + 0.529·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 714025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 714025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(714025\)    =    \(5^{2} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(45.5268\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 714025,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.526233200\)
\(L(\frac12)\) \(\approx\) \(2.526233200\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad5$C_1$ \( ( 1 - T )^{2} \)
13 \( 1 \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.2.a_b
3$D_{4}$ \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.3.ac_e
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.7.e_s
11$D_{4}$ \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.11.ag_bc
17$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.17.a_w
19$D_{4}$ \( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.19.ac_m
23$D_{4}$ \( 1 - 6 T + 52 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_ca
29$D_{4}$ \( 1 + 12 T + 82 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.29.m_de
31$D_{4}$ \( 1 + 10 T + 60 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.31.k_ci
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.37.ai_dm
41$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.41.a_cs
43$D_{4}$ \( 1 - 10 T + 84 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.43.ak_dg
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.47.m_fa
53$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.53.a_ac
59$D_{4}$ \( 1 - 6 T - 20 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.59.ag_au
61$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.61.ae_s
67$D_{4}$ \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.67.ai_bq
71$D_{4}$ \( 1 + 6 T + 148 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.71.g_fs
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.73.ai_gg
79$D_{4}$ \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.79.ae_cc
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.83.am_hu
89$D_{4}$ \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.89.am_gk
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.97.e_hq
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

−9.928623450692665846012042566051, −9.804255308650575402490660463028, −9.415849134480595536132715547200, −9.180506015945554642003228424302, −8.887225643326998679649085215482, −8.705903813127830158769042037016, −7.81884854997300313008022268477, −7.46185351023504673670762954925, −6.81816797969766121577000051320, −6.66640281247935516679368286505, −6.03255528736792553513960384729, −5.75652089371752660746428406732, −5.05886021189905239270894164588, −4.52562817767662100232848733866, −3.65059913020362126207314055427, −3.57336729733123206830438464359, −3.10892688779358743516607554304, −2.26526449119312504336081390871, −1.85773614721349459986607109893, −0.73029408476278882416789874489, 0.73029408476278882416789874489, 1.85773614721349459986607109893, 2.26526449119312504336081390871, 3.10892688779358743516607554304, 3.57336729733123206830438464359, 3.65059913020362126207314055427, 4.52562817767662100232848733866, 5.05886021189905239270894164588, 5.75652089371752660746428406732, 6.03255528736792553513960384729, 6.66640281247935516679368286505, 6.81816797969766121577000051320, 7.46185351023504673670762954925, 7.81884854997300313008022268477, 8.705903813127830158769042037016, 8.887225643326998679649085215482, 9.180506015945554642003228424302, 9.415849134480595536132715547200, 9.804255308650575402490660463028, 9.928623450692665846012042566051

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