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Properties

Label 4-4998e2-1.1-c1e2-0-11
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 + 4·11-s − 6·12-s − 5·13-s + 4·15-s + 5·16-s − 2·17-s + 6·18-s − 5·19-s − 6·20-s + 8·22-s + 3·23-s − 8·24-s − 7·25-s − 10·26-s − 4·27-s + 29-s + 8·30-s + 5·31-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 + 1.20·11-s − 1.73·12-s − 1.38·13-s + 1.03·15-s + 5/4·16-s − 0.485·17-s + 1.41·18-s − 1.14·19-s − 1.34·20-s + 1.70·22-s + 0.625·23-s − 1.63·24-s − 7/5·25-s − 1.96·26-s − 0.769·27-s + 0.185·29-s + 1.46·30-s + 0.898·31-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$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.11.ae_ba
13$D_{4}$ \( 1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.13.f_s
19$D_{4}$ \( 1 + 5 T + 30 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.19.f_be
23$D_{4}$ \( 1 - 3 T + 34 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.23.ad_bi
29$D_{4}$ \( 1 - T + 44 T^{2} - p T^{3} + p^{2} T^{4} \) 2.29.ab_bs
31$D_{4}$ \( 1 - 5 T + 54 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.31.af_cc
37$D_{4}$ \( 1 + T - 54 T^{2} + p T^{3} + p^{2} T^{4} \) 2.37.b_acc
41$D_{4}$ \( 1 + 5 T + 74 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.41.f_cw
43$D_{4}$ \( 1 - 3 T + 74 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.43.ad_cw
47$D_{4}$ \( 1 + 15 T + 136 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.47.p_fg
53$D_{4}$ \( 1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.53.g_cg
59$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.59.ag_ex
61$D_{4}$ \( 1 + 10 T + 90 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.61.k_dm
67$D_{4}$ \( 1 + 3 T + 122 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.67.d_es
71$D_{4}$ \( 1 - 3 T + 130 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.71.ad_fa
73$D_{4}$ \( 1 + 2 T + 90 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.73.c_dm
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.79.q_io
83$D_{4}$ \( 1 - 13 T + 194 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.83.an_hm
89$D_{4}$ \( 1 + 15 T + 220 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.89.p_im
97$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.97.i_ic
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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

−7.80553731221092161736101038784, −7.65124005659471905068652585207, −6.96141060981355009594076664449, −6.94767360174576603170829651775, −6.56665513803449115755386033022, −6.25737915981208275894897080588, −5.85824373761924750365317519153, −5.50861034608516998560593206065, −4.89504495972828683496442849321, −4.80545669962910999952070042008, −4.36402789892407703298504213406, −4.16222613775652327310506642982, −3.63964127537337705721429215041, −3.39279503661334156866489762140, −2.55841475188646522550125605043, −2.42484832377762571397778137287, −1.46466726180295791510570889793, −1.41480436658942971360118706920, 0, 0, 1.41480436658942971360118706920, 1.46466726180295791510570889793, 2.42484832377762571397778137287, 2.55841475188646522550125605043, 3.39279503661334156866489762140, 3.63964127537337705721429215041, 4.16222613775652327310506642982, 4.36402789892407703298504213406, 4.80545669962910999952070042008, 4.89504495972828683496442849321, 5.50861034608516998560593206065, 5.85824373761924750365317519153, 6.25737915981208275894897080588, 6.56665513803449115755386033022, 6.94767360174576603170829651775, 6.96141060981355009594076664449, 7.65124005659471905068652585207, 7.80553731221092161736101038784

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