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Properties

Label 4-1690e2-1.1-c1e2-0-13
Degree $4$
Conductor $2856100$
Sign $1$
Analytic cond. $182.107$
Root an. cond. $3.67351$
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 + 2·3-s + 3·4-s + 2·5-s − 4·6-s − 2·7-s − 4·8-s − 4·10-s + 6·12-s + 4·14-s + 4·15-s + 5·16-s + 6·17-s + 4·19-s + 6·20-s − 4·21-s − 8·24-s + 3·25-s − 2·27-s − 6·28-s + 12·29-s − 8·30-s + 10·31-s − 6·32-s − 12·34-s − 4·35-s − 8·37-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.894·5-s − 1.63·6-s − 0.755·7-s − 1.41·8-s − 1.26·10-s + 1.73·12-s + 1.06·14-s + 1.03·15-s + 5/4·16-s + 1.45·17-s + 0.917·19-s + 1.34·20-s − 0.872·21-s − 1.63·24-s + 3/5·25-s − 0.384·27-s − 1.13·28-s + 2.22·29-s − 1.46·30-s + 1.79·31-s − 1.06·32-s − 2.05·34-s − 0.676·35-s − 1.31·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2856100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2856100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2856100\)    =    \(2^{2} \cdot 5^{2} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(182.107\)
Root analytic conductor: \(3.67351\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2856100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.465149940\)
\(L(\frac12)\) \(\approx\) \(2.465149940\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
5$C_1$ \( ( 1 - T )^{2} \)
13 \( 1 \)
good3$D_{4}$ \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.3.ac_e
7$D_{4}$ \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_d
11$C_2^2$ \( 1 + 19 T^{2} + p^{2} T^{4} \) 2.11.a_t
17$D_{4}$ \( 1 - 6 T + 40 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.17.ag_bo
19$D_{4}$ \( 1 - 4 T + 15 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.19.ae_p
23$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.23.a_bi
29$D_{4}$ \( 1 - 12 T + 82 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.29.am_de
31$D_{4}$ \( 1 - 10 T + 84 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.31.ak_dg
37$D_{4}$ \( 1 + 8 T + 63 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.37.i_cl
41$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.41.a_bi
43$D_{4}$ \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.43.ae_bq
47$D_{4}$ \( 1 + 6 T + 91 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.47.g_dn
53$C_2^2$ \( 1 + 103 T^{2} + p^{2} T^{4} \) 2.53.a_dz
59$D_{4}$ \( 1 - 12 T + 142 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.59.am_fm
61$D_{4}$ \( 1 - 10 T + 144 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.61.ak_fo
67$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.67.ae_be
71$D_{4}$ \( 1 - 24 T + 274 T^{2} - 24 p T^{3} + p^{2} T^{4} \) 2.71.ay_ko
73$D_{4}$ \( 1 + 2 T + 144 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.73.c_fo
79$D_{4}$ \( 1 - 10 T + 180 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.79.ak_gy
83$D_{4}$ \( 1 - 6 T + 100 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.83.ag_dw
89$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.89.s_jz
97$D_{4}$ \( 1 + 14 T + 240 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.97.o_jg
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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

−9.589239195906294047567217020776, −9.261413895551029949290330779197, −8.610057147488264148816380916667, −8.455699527783633237975256262287, −8.046453929914817214974844208946, −7.995267809897415304840265387924, −7.07927035522650330657652585427, −6.95596430629141929750123182739, −6.47905470177036460954364467784, −6.17607752752829745208326939956, −5.43614878127168211271144919930, −5.33442112178138922812138215387, −4.62304220427540059167546617309, −3.74858312957131062066192165814, −3.21213249062760426348717493072, −3.05796207421701663304036738759, −2.45148839577455265609093602291, −2.14016745913155624963032018499, −1.11247019135955241962420012521, −0.843142401467970765558933618765, 0.843142401467970765558933618765, 1.11247019135955241962420012521, 2.14016745913155624963032018499, 2.45148839577455265609093602291, 3.05796207421701663304036738759, 3.21213249062760426348717493072, 3.74858312957131062066192165814, 4.62304220427540059167546617309, 5.33442112178138922812138215387, 5.43614878127168211271144919930, 6.17607752752829745208326939956, 6.47905470177036460954364467784, 6.95596430629141929750123182739, 7.07927035522650330657652585427, 7.995267809897415304840265387924, 8.046453929914817214974844208946, 8.455699527783633237975256262287, 8.610057147488264148816380916667, 9.261413895551029949290330779197, 9.589239195906294047567217020776

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