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Properties

Label 4-5054e2-1.1-c1e2-0-9
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 − 3·3-s + 3·4-s − 4·5-s − 6·6-s − 2·7-s + 4·8-s + 2·9-s − 8·10-s + 6·11-s − 9·12-s + 13-s − 4·14-s + 12·15-s + 5·16-s − 3·17-s + 4·18-s − 12·20-s + 6·21-s + 12·22-s − 12·24-s + 7·25-s + 2·26-s + 6·27-s − 6·28-s + 12·29-s + 24·30-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.73·3-s + 3/2·4-s − 1.78·5-s − 2.44·6-s − 0.755·7-s + 1.41·8-s + 2/3·9-s − 2.52·10-s + 1.80·11-s − 2.59·12-s + 0.277·13-s − 1.06·14-s + 3.09·15-s + 5/4·16-s − 0.727·17-s + 0.942·18-s − 2.68·20-s + 1.30·21-s + 2.55·22-s − 2.44·24-s + 7/5·25-s + 0.392·26-s + 1.15·27-s − 1.13·28-s + 2.22·29-s + 4.38·30-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$D_{4}$ \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) 2.3.d_h
5$D_{4}$ \( 1 + 4 T + 9 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.5.e_j
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.11.ag_bf
13$D_{4}$ \( 1 - T + 15 T^{2} - p T^{3} + p^{2} T^{4} \) 2.13.ab_p
17$D_{4}$ \( 1 + 3 T + 25 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.17.d_z
23$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.23.a_b
29$D_{4}$ \( 1 - 12 T + 89 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.29.am_dl
31$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.31.k_dj
37$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.37.ak_dv
41$D_{4}$ \( 1 + 3 T + 53 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.41.d_cb
43$D_{4}$ \( 1 + 3 T - 13 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.43.d_an
47$D_{4}$ \( 1 + 8 T + 105 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.47.i_eb
53$D_{4}$ \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.53.am_dt
59$D_{4}$ \( 1 + 16 T + 3 p T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.59.q_gv
61$C_2^2$ \( 1 + 77 T^{2} + p^{2} T^{4} \) 2.61.a_cz
67$D_{4}$ \( 1 + 11 T + 63 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.67.l_cl
71$D_{4}$ \( 1 + 16 T + 186 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.71.q_he
73$D_{4}$ \( 1 + 7 T + 127 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.73.h_ex
79$D_{4}$ \( 1 + 27 T + 329 T^{2} + 27 p T^{3} + p^{2} T^{4} \) 2.79.bb_mr
83$D_{4}$ \( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.83.e_fu
89$D_{4}$ \( 1 - 5 T + 183 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.89.af_hb
97$D_{4}$ \( 1 - 24 T + 293 T^{2} - 24 p T^{3} + p^{2} T^{4} \) 2.97.ay_lh
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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

−7.935231853611193153903462142729, −7.30327189351335576514860949510, −7.04309238014470053871464462233, −7.04005691000821066696607345712, −6.36959955823973951078716651942, −6.12546907394498046253238757389, −5.90055911330653196652715766193, −5.78340018747528314465134444622, −4.78498716993773470802334171478, −4.71849670704738387897739752896, −4.55261144070069790623322491264, −4.02668717735028189981736651053, −3.50737109255495899081237430264, −3.49457823241203183224752609022, −2.84421019872987466891801361875, −2.41656594976030202015142720759, −1.26717996519117407440489193144, −1.23444376252803623352156962715, 0, 0, 1.23444376252803623352156962715, 1.26717996519117407440489193144, 2.41656594976030202015142720759, 2.84421019872987466891801361875, 3.49457823241203183224752609022, 3.50737109255495899081237430264, 4.02668717735028189981736651053, 4.55261144070069790623322491264, 4.71849670704738387897739752896, 4.78498716993773470802334171478, 5.78340018747528314465134444622, 5.90055911330653196652715766193, 6.12546907394498046253238757389, 6.36959955823973951078716651942, 7.04005691000821066696607345712, 7.04309238014470053871464462233, 7.30327189351335576514860949510, 7.935231853611193153903462142729

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