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Properties

Label 4-630e2-1.1-c1e2-0-17
Degree $4$
Conductor $396900$
Sign $1$
Analytic cond. $25.3066$
Root an. cond. $2.24289$
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-s − 3·3-s + 5-s − 3·6-s − 7-s − 8-s + 6·9-s + 10-s − 3·11-s + 4·13-s − 14-s − 3·15-s − 16-s + 6·17-s + 6·18-s − 14·19-s + 3·21-s − 3·22-s + 6·23-s + 3·24-s + 4·26-s − 9·27-s + 6·29-s − 3·30-s + 10·31-s + 9·33-s + 6·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.73·3-s + 0.447·5-s − 1.22·6-s − 0.377·7-s − 0.353·8-s + 2·9-s + 0.316·10-s − 0.904·11-s + 1.10·13-s − 0.267·14-s − 0.774·15-s − 1/4·16-s + 1.45·17-s + 1.41·18-s − 3.21·19-s + 0.654·21-s − 0.639·22-s + 1.25·23-s + 0.612·24-s + 0.784·26-s − 1.73·27-s + 1.11·29-s − 0.547·30-s + 1.79·31-s + 1.56·33-s + 1.02·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(396900\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(25.3066\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 396900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.231170499\)
\(L(\frac12)\) \(\approx\) \(1.231170499\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3$C_2$ \( 1 + p T + p T^{2} \)
5$C_2$ \( 1 - T + T^{2} \)
7$C_2$ \( 1 + T + T^{2} \)
good11$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.11.d_ac
13$C_2^2$ \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.13.ae_d
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.17.ag_br
19$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.19.o_dj
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_n
29$C_2^2$ \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.29.ag_h
31$C_2^2$ \( 1 - 10 T + 69 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.31.ak_cr
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.37.aq_fi
41$C_2^2$ \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.41.j_bo
43$C_2^2$ \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) 2.43.ab_abq
47$C_2^2$ \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.47.m_dt
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2^2$ \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.59.j_w
61$C_2$ \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.61.o_ff
67$C_2^2$ \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.67.ah_as
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.71.am_gw
73$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.73.o_hn
79$C_2^2$ \( 1 + 2 T - 75 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.79.c_acx
83$C_2^2$ \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.83.am_cj
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_ig
97$C_2^2$ \( 1 + 11 T + 24 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.97.l_y
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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

−10.69807178842631840647451144202, −10.56760135807349868326809112196, −10.13741094357174451878621281532, −9.829016613268438883221680717459, −9.120230554505570598888224331785, −8.628840764224520108233239347229, −7.984796346894655846101931496928, −7.909870642544675970084118990727, −6.77331916281562260774049178943, −6.55663842289685601167195634305, −6.15687998807320618404880839805, −6.04943749075796288599573865604, −5.11816319664591450367153839155, −5.11275081329142052909592205404, −4.26150984347762003833465096269, −4.17347104829828329042001660538, −3.08904956914175562871354698319, −2.63094255487437300804473523083, −1.52114380416761937174305248058, −0.63835042020437344778976500790, 0.63835042020437344778976500790, 1.52114380416761937174305248058, 2.63094255487437300804473523083, 3.08904956914175562871354698319, 4.17347104829828329042001660538, 4.26150984347762003833465096269, 5.11275081329142052909592205404, 5.11816319664591450367153839155, 6.04943749075796288599573865604, 6.15687998807320618404880839805, 6.55663842289685601167195634305, 6.77331916281562260774049178943, 7.909870642544675970084118990727, 7.984796346894655846101931496928, 8.628840764224520108233239347229, 9.120230554505570598888224331785, 9.829016613268438883221680717459, 10.13741094357174451878621281532, 10.56760135807349868326809112196, 10.69807178842631840647451144202

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