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Properties

Label 4-4998e2-1.1-c1e2-0-13
Degree $4$
Conductor $24980004$
Sign $1$
Analytic cond. $1592.74$
Root an. cond. $6.31737$
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 + 2·3-s + 3·4-s − 2·5-s + 4·6-s + 4·8-s + 3·9-s − 4·10-s − 6·11-s + 6·12-s + 2·13-s − 4·15-s + 5·16-s − 2·17-s + 6·18-s − 4·19-s − 6·20-s − 12·22-s − 12·23-s + 8·24-s − 7·25-s + 4·26-s + 4·27-s − 8·29-s − 8·30-s − 8·31-s + 6·32-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 3/2·4-s − 0.894·5-s + 1.63·6-s + 1.41·8-s + 9-s − 1.26·10-s − 1.80·11-s + 1.73·12-s + 0.554·13-s − 1.03·15-s + 5/4·16-s − 0.485·17-s + 1.41·18-s − 0.917·19-s − 1.34·20-s − 2.55·22-s − 2.50·23-s + 1.63·24-s − 7/5·25-s + 0.784·26-s + 0.769·27-s − 1.48·29-s − 1.46·30-s − 1.43·31-s + 1.06·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24980004 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24980004 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(24980004\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{4} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(1592.74\)
Root analytic conductor: \(6.31737\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 24980004,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 - T )^{2} \)
3$C_1$ \( ( 1 - T )^{2} \)
7 \( 1 \)
17$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.5.c_l
11$D_{4}$ \( 1 + 6 T + 29 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.11.g_bd
13$D_{4}$ \( 1 - 2 T + 19 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.13.ac_t
19$D_{4}$ \( 1 + 4 T + 24 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.19.e_y
23$C_4$ \( 1 + 12 T + 74 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.23.m_cw
29$D_{4}$ \( 1 + 8 T + 72 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.29.i_cu
31$D_{4}$ \( 1 + 8 T + 60 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.31.i_ci
37$D_{4}$ \( 1 + 10 T + 67 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.37.k_cp
41$D_{4}$ \( 1 + 4 T + 68 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.41.e_cq
43$D_{4}$ \( 1 + 10 T + 109 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.43.k_ef
47$D_{4}$ \( 1 - 16 T + 140 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.47.aq_fk
53$D_{4}$ \( 1 - 2 T + 99 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.53.ac_dv
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.59.ai_fe
61$D_{4}$ \( 1 + 12 T + 126 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.61.m_ew
67$D_{4}$ \( 1 + 10 T + 61 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.67.k_cj
71$D_{4}$ \( 1 - 8 T - 4 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.71.ai_ae
73$D_{4}$ \( 1 + 14 T + 163 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.73.o_gh
79$D_{4}$ \( 1 + 10 T + 181 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.79.k_gz
83$D_{4}$ \( 1 - 14 T + 213 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.83.ao_if
89$D_{4}$ \( 1 - 22 T + 3 p T^{2} - 22 p T^{3} + p^{2} T^{4} \) 2.89.aw_kh
97$D_{4}$ \( 1 - 6 T + 195 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.97.ag_hn
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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

−7.77241981987145449091893512422, −7.71700967778242078328882801199, −7.33962148830436663522419513203, −7.32847563892490985226440719407, −6.42546089759009161672579771979, −6.26477940748208019091426593596, −5.75930150876437789436849885043, −5.56093579462307517111194394751, −4.92083148830082758503110446409, −4.79373453639274746404688080022, −4.01544551245912146558098220824, −3.92115061610559210597525556062, −3.55013006670295957174203009908, −3.48658817820894353302422736919, −2.47569177749731779764648823034, −2.45252851567476973510570487421, −1.77072245249426437287142844679, −1.75961639239454982073767736550, 0, 0, 1.75961639239454982073767736550, 1.77072245249426437287142844679, 2.45252851567476973510570487421, 2.47569177749731779764648823034, 3.48658817820894353302422736919, 3.55013006670295957174203009908, 3.92115061610559210597525556062, 4.01544551245912146558098220824, 4.79373453639274746404688080022, 4.92083148830082758503110446409, 5.56093579462307517111194394751, 5.75930150876437789436849885043, 6.26477940748208019091426593596, 6.42546089759009161672579771979, 7.32847563892490985226440719407, 7.33962148830436663522419513203, 7.71700967778242078328882801199, 7.77241981987145449091893512422

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