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Properties

Label 4-2646e2-1.1-c1e2-0-26
Degree $4$
Conductor $7001316$
Sign $1$
Analytic cond. $446.409$
Root an. cond. $4.59656$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·4-s + 4·8-s − 4·11-s + 6·13-s + 5·16-s + 6·17-s + 12·19-s − 8·22-s − 2·23-s − 8·25-s + 12·26-s − 2·29-s + 6·31-s + 6·32-s + 12·34-s − 4·37-s + 24·38-s − 2·43-s − 12·44-s − 4·46-s − 16·50-s + 18·52-s + 14·53-s − 4·58-s + 6·59-s + 12·61-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 1.41·8-s − 1.20·11-s + 1.66·13-s + 5/4·16-s + 1.45·17-s + 2.75·19-s − 1.70·22-s − 0.417·23-s − 8/5·25-s + 2.35·26-s − 0.371·29-s + 1.07·31-s + 1.06·32-s + 2.05·34-s − 0.657·37-s + 3.89·38-s − 0.304·43-s − 1.80·44-s − 0.589·46-s − 2.26·50-s + 2.49·52-s + 1.92·53-s − 0.525·58-s + 0.781·59-s + 1.53·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7001316 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7001316 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7001316\)    =    \(2^{2} \cdot 3^{6} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(446.409\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7001316,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(9.022405760\)
\(L(\frac12)\) \(\approx\) \(9.022405760\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.5.a_i
11$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.11.e_i
13$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.13.ag_bj
17$D_{4}$ \( 1 - 6 T + 41 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.17.ag_bp
19$C_2^2$ \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.19.am_cu
23$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.23.c_bv
29$D_{4}$ \( 1 + 2 T + 41 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.29.c_bp
31$D_{4}$ \( 1 - 6 T + 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.31.ag_cb
37$D_{4}$ \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.37.e_g
41$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \) 2.41.a_cw
43$D_{4}$ \( 1 + 2 T + 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.43.c_cr
47$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \) 2.47.a_bs
53$D_{4}$ \( 1 - 14 T + 137 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.53.ao_fh
59$D_{4}$ \( 1 - 6 T + 119 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.59.ag_ep
61$D_{4}$ \( 1 - 12 T + 126 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.61.am_ew
67$D_{4}$ \( 1 - 18 T + 197 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.67.as_hp
71$D_{4}$ \( 1 - 6 T + 79 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.71.ag_db
73$C_2^2$ \( 1 + 48 T^{2} + p^{2} T^{4} \) 2.73.a_bw
79$D_{4}$ \( 1 - 4 T - 4 p T^{3} + p^{2} T^{4} \) 2.79.ae_a
83$D_{4}$ \( 1 - 12 T + 74 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.83.am_cw
89$D_{4}$ \( 1 + 18 T + 209 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.89.s_ib
97$D_{4}$ \( 1 - 24 T + 330 T^{2} - 24 p T^{3} + p^{2} T^{4} \) 2.97.ay_ms
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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

−8.904402627740010420481192853669, −8.622644129773862539529832504706, −8.012897113797102669409786590718, −7.79379376956131483533436271788, −7.59407855485482138496048055403, −7.17410808353620848711608687012, −6.43660470973841071689009844649, −6.43294557206771289998648121994, −5.58889081562060924951492103563, −5.58091837216949866736706019275, −5.21895273110017556064096732903, −5.02328257322575875980958178394, −4.05524389291606364540711530746, −3.82454224164207829957955491269, −3.45810404312383599362920006109, −3.19336032846081947983380085263, −2.46674086760496643756866727522, −2.12787860985199687341077284804, −1.20995913547507016240539929411, −0.890384696189987934251088519419, 0.890384696189987934251088519419, 1.20995913547507016240539929411, 2.12787860985199687341077284804, 2.46674086760496643756866727522, 3.19336032846081947983380085263, 3.45810404312383599362920006109, 3.82454224164207829957955491269, 4.05524389291606364540711530746, 5.02328257322575875980958178394, 5.21895273110017556064096732903, 5.58091837216949866736706019275, 5.58889081562060924951492103563, 6.43294557206771289998648121994, 6.43660470973841071689009844649, 7.17410808353620848711608687012, 7.59407855485482138496048055403, 7.79379376956131483533436271788, 8.012897113797102669409786590718, 8.622644129773862539529832504706, 8.904402627740010420481192853669

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