↑

Properties

Label 4-65e4-1.1-c1e2-0-3
Degree $4$
Conductor $17850625$
Sign $1$
Analytic cond. $1138.17$
Root an. cond. $5.80833$
Motivic weight $1$
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  − 2-s − 2·4-s − 4·7-s + 3·8-s − 9-s + 4·11-s + 4·14-s + 16-s − 2·17-s + 18-s + 4·19-s − 4·22-s − 12·23-s + 8·28-s − 6·29-s − 2·32-s + 2·34-s + 2·36-s − 6·37-s − 4·38-s + 6·41-s + 8·43-s − 8·44-s + 12·46-s − 8·47-s + 3·49-s − 12·53-s + ⋯
L(s)  = 1  − 0.707·2-s − 4-s − 1.51·7-s + 1.06·8-s − 1/3·9-s + 1.20·11-s + 1.06·14-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.917·19-s − 0.852·22-s − 2.50·23-s + 1.51·28-s − 1.11·29-s − 0.353·32-s + 0.342·34-s + 1/3·36-s − 0.986·37-s − 0.648·38-s + 0.937·41-s + 1.21·43-s − 1.20·44-s + 1.76·46-s − 1.16·47-s + 3/7·49-s − 1.64·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17850625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17850625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(17850625\)    =    \(5^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(1138.17\)
Root analytic conductor: \(5.80833\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 17850625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 \)
good2$D_{4}$ \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) 2.2.b_d
3$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.3.a_b
7$D_{4}$ \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.7.e_n
11$D_{4}$ \( 1 - 4 T + 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.11.ae_v
17$D_{4}$ \( 1 + 2 T + 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.17.c_p
19$D_{4}$ \( 1 - 4 T + 37 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.19.ae_bl
23$D_{4}$ \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.23.m_cz
29$D_{4}$ \( 1 + 6 T + 47 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.29.g_bv
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.37.g_df
41$D_{4}$ \( 1 - 6 T + 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.41.ag_l
43$D_{4}$ \( 1 - 8 T + 97 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.43.ai_dt
47$D_{4}$ \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.47.i_be
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.53.m_fm
59$D_{4}$ \( 1 - 12 T + 109 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.59.am_ef
61$C_2^2$ \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.61.ac_acf
67$D_{4}$ \( 1 + 8 T + 105 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.67.i_eb
71$D_{4}$ \( 1 + 8 T + 153 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.71.i_fx
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.73.am_ha
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \) 2.83.a_di
89$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.89.s_jz
97$D_{4}$ \( 1 + 2 T + 175 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.97.c_gt
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

−8.347335800920543497321056727103, −7.891020470638358748050442508901, −7.66216578485817698630883305911, −7.10333643785448412046456784863, −6.82648345573417059571305761680, −6.24997722291685600076052178965, −6.19868354538622317846590729564, −5.65345086848535316835190627288, −5.39738957406433613297415107109, −4.76320314526641552223380149261, −4.22515343568829924969986309058, −4.09344021881331229013426994023, −3.61029668018814669742955620150, −3.30252252402054877417439070975, −2.74936908844570000786376049919, −2.08057816170988343724375407856, −1.59035537353730054684932430914, −0.909849145766315883704496109797, 0, 0, 0.909849145766315883704496109797, 1.59035537353730054684932430914, 2.08057816170988343724375407856, 2.74936908844570000786376049919, 3.30252252402054877417439070975, 3.61029668018814669742955620150, 4.09344021881331229013426994023, 4.22515343568829924969986309058, 4.76320314526641552223380149261, 5.39738957406433613297415107109, 5.65345086848535316835190627288, 6.19868354538622317846590729564, 6.24997722291685600076052178965, 6.82648345573417059571305761680, 7.10333643785448412046456784863, 7.66216578485817698630883305911, 7.891020470638358748050442508901, 8.347335800920543497321056727103

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