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Properties

Label 4-2646e2-1.1-c1e2-0-15
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s + 3·5-s − 4·8-s − 6·10-s − 3·11-s + 5·13-s + 5·16-s − 3·17-s + 5·19-s + 9·20-s + 6·22-s − 3·23-s + 5·25-s − 10·26-s − 3·29-s + 8·31-s − 6·32-s + 6·34-s + 7·37-s − 10·38-s − 12·40-s + 9·41-s − 11·43-s − 9·44-s + 6·46-s − 10·50-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s + 1.34·5-s − 1.41·8-s − 1.89·10-s − 0.904·11-s + 1.38·13-s + 5/4·16-s − 0.727·17-s + 1.14·19-s + 2.01·20-s + 1.27·22-s − 0.625·23-s + 25-s − 1.96·26-s − 0.557·29-s + 1.43·31-s − 1.06·32-s + 1.02·34-s + 1.15·37-s − 1.62·38-s − 1.89·40-s + 1.40·41-s − 1.67·43-s − 1.35·44-s + 0.884·46-s − 1.41·50-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

\(L(1)\) \(\approx\) \(1.822570033\)
\(L(\frac12)\) \(\approx\) \(1.822570033\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.5.ad_e
11$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.11.d_ac
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.af_m
17$C_2^2$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.17.d_ai
19$C_2^2$ \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.19.af_g
23$C_2^2$ \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.23.d_ao
29$C_2^2$ \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.29.d_au
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.31.ai_da
37$C_2^2$ \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.37.ah_m
41$C_2^2$ \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.41.aj_bo
43$C_2^2$ \( 1 + 11 T + 78 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.43.l_da
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2^2$ \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.53.d_abs
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.59.ay_kc
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.61.e_ew
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.67.i_fu
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2^2$ \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.73.al_bw
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.79.aq_io
83$C_2^2$ \( 1 + 3 T - 74 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.83.d_acw
89$C_2^2$ \( 1 + 15 T + 136 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.89.p_fg
97$C_2^2$ \( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} \) 2.97.b_ads
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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

−8.868667717466868427297937666111, −8.856887158414794940244571787048, −8.338584600625961424429941979486, −7.997601305115960714750129300333, −7.71364139382464663298806198455, −7.23308592188714657938947376942, −6.73394335083558953687109286973, −6.42391657737435983604459445735, −6.02994354463711447345906345660, −5.85218606543768584543169184627, −5.11846496603075847098559510429, −5.11751060941928978195390488187, −4.19045128694439060141462000973, −3.77244562972407106360213876243, −3.09199679851013379680095574931, −2.66804047836263194591353122262, −2.26176015624340639066544856453, −1.74606248370178456779252796391, −1.15571204934586437826151436294, −0.61750858275570578847911970614, 0.61750858275570578847911970614, 1.15571204934586437826151436294, 1.74606248370178456779252796391, 2.26176015624340639066544856453, 2.66804047836263194591353122262, 3.09199679851013379680095574931, 3.77244562972407106360213876243, 4.19045128694439060141462000973, 5.11751060941928978195390488187, 5.11846496603075847098559510429, 5.85218606543768584543169184627, 6.02994354463711447345906345660, 6.42391657737435983604459445735, 6.73394335083558953687109286973, 7.23308592188714657938947376942, 7.71364139382464663298806198455, 7.997601305115960714750129300333, 8.338584600625961424429941979486, 8.856887158414794940244571787048, 8.868667717466868427297937666111

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