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Properties

Label 4-5054e2-1.1-c1e2-0-12
Degree $4$
Conductor $25542916$
Sign $1$
Analytic cond. $1628.63$
Root an. cond. $6.35266$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3-s + 3·4-s + 2·5-s + 2·6-s − 2·7-s + 4·8-s − 4·9-s + 4·10-s − 10·11-s + 3·12-s + 3·13-s − 4·14-s + 2·15-s + 5·16-s + 7·17-s − 8·18-s + 6·20-s − 2·21-s − 20·22-s − 12·23-s + 4·24-s − 7·25-s + 6·26-s − 6·27-s − 6·28-s − 12·29-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.577·3-s + 3/2·4-s + 0.894·5-s + 0.816·6-s − 0.755·7-s + 1.41·8-s − 4/3·9-s + 1.26·10-s − 3.01·11-s + 0.866·12-s + 0.832·13-s − 1.06·14-s + 0.516·15-s + 5/4·16-s + 1.69·17-s − 1.88·18-s + 1.34·20-s − 0.436·21-s − 4.26·22-s − 2.50·23-s + 0.816·24-s − 7/5·25-s + 1.17·26-s − 1.15·27-s − 1.13·28-s − 2.22·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25542916 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25542916 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(25542916\)    =    \(2^{2} \cdot 7^{2} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(1628.63\)
Root analytic conductor: \(6.35266\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 25542916,\ (\ :1/2, 1/2),\ 1)\)

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 $\mathbf{F}_p$
bad2$C_1$ \( ( 1 - T )^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
19 \( 1 \)
good3$D_{4}$ \( 1 - T + 5 T^{2} - p T^{3} + p^{2} T^{4} \) 2.3.ab_f
5$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.5.ac_l
11$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.11.k_bv
13$D_{4}$ \( 1 - 3 T + 17 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.13.ad_r
17$D_{4}$ \( 1 - 7 T + 35 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.17.ah_bj
23$D_{4}$ \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.23.m_cz
29$D_{4}$ \( 1 + 12 T + 89 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.29.m_dl
31$C_2^2$ \( 1 + 17 T^{2} + p^{2} T^{4} \) 2.31.a_r
37$D_{4}$ \( 1 + 6 T + 3 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.37.g_d
41$C_4$ \( 1 - 9 T + 71 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.41.aj_ct
43$D_{4}$ \( 1 + 7 T + 87 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.43.h_dj
47$D_{4}$ \( 1 + 2 T + 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.47.c_p
53$D_{4}$ \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.53.m_dt
59$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.59.aw_jf
61$D_{4}$ \( 1 - 6 T + 111 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.61.ag_eh
67$D_{4}$ \( 1 - 9 T + 143 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.67.aj_fn
71$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.71.u_ji
73$D_{4}$ \( 1 + 9 T + 65 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.73.j_cn
79$D_{4}$ \( 1 + 11 T + 177 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.79.l_gv
83$D_{4}$ \( 1 + 4 T - 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.83.e_ak
89$D_{4}$ \( 1 - 3 T + 169 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.89.ad_gn
97$D_{4}$ \( 1 + 10 T + 199 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.97.k_hr
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.894485621678093162929055563888, −7.88715610108539321239776415825, −7.28302754501355177556202419465, −7.06845957627723859894693942724, −6.24310103125100703018274333698, −5.96780323568502429672394527833, −5.67776377120205288220024769371, −5.66270192552395995127594064525, −5.34975056671965840447134039218, −4.98961717681091109412073528918, −4.02620808779089183486358775718, −3.98159073182168572648894385208, −3.34746484326330225806900423569, −3.24512899773144223555405793387, −2.58307551410786850409947326964, −2.44152619398637446730108803534, −1.98765026880988300524314540212, −1.52007451308014621548898381245, 0, 0, 1.52007451308014621548898381245, 1.98765026880988300524314540212, 2.44152619398637446730108803534, 2.58307551410786850409947326964, 3.24512899773144223555405793387, 3.34746484326330225806900423569, 3.98159073182168572648894385208, 4.02620808779089183486358775718, 4.98961717681091109412073528918, 5.34975056671965840447134039218, 5.66270192552395995127594064525, 5.67776377120205288220024769371, 5.96780323568502429672394527833, 6.24310103125100703018274333698, 7.06845957627723859894693942724, 7.28302754501355177556202419465, 7.88715610108539321239776415825, 7.894485621678093162929055563888

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