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Properties

Label 4-440e2-1.1-c1e2-0-9
Degree $4$
Conductor $193600$
Sign $1$
Analytic cond. $12.3441$
Root an. cond. $1.87441$
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  + 3-s − 2·5-s + 5·7-s − 9-s − 2·11-s + 6·13-s − 2·15-s − 3·17-s + 7·19-s + 5·21-s − 6·23-s + 3·25-s + 13·29-s − 7·31-s − 2·33-s − 10·35-s + 19·37-s + 6·39-s − 8·41-s + 2·43-s + 2·45-s + 2·47-s + 9·49-s − 3·51-s + 7·53-s + 4·55-s + 7·57-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s + 1.88·7-s − 1/3·9-s − 0.603·11-s + 1.66·13-s − 0.516·15-s − 0.727·17-s + 1.60·19-s + 1.09·21-s − 1.25·23-s + 3/5·25-s + 2.41·29-s − 1.25·31-s − 0.348·33-s − 1.69·35-s + 3.12·37-s + 0.960·39-s − 1.24·41-s + 0.304·43-s + 0.298·45-s + 0.291·47-s + 9/7·49-s − 0.420·51-s + 0.961·53-s + 0.539·55-s + 0.927·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(193600\)    =    \(2^{6} \cdot 5^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(12.3441\)
Root analytic conductor: \(1.87441\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 193600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.281885977\)
\(L(\frac12)\) \(\approx\) \(2.281885977\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
11$C_1$ \( ( 1 + T )^{2} \)
good3$D_{4}$ \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) 2.3.ab_c
7$D_{4}$ \( 1 - 5 T + 16 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.7.af_q
13$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.13.ag_s
17$D_{4}$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.17.d_ac
19$C_4$ \( 1 - 7 T + 46 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.19.ah_bu
23$D_{4}$ \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.23.g_bm
29$D_{4}$ \( 1 - 13 T + 96 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.29.an_ds
31$D_{4}$ \( 1 + 7 T + 70 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.31.h_cs
37$D_{4}$ \( 1 - 19 T + 160 T^{2} - 19 p T^{3} + p^{2} T^{4} \) 2.37.at_ge
41$D_{4}$ \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.41.i_be
43$D_{4}$ \( 1 - 2 T + 70 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.43.ac_cs
47$D_{4}$ \( 1 - 2 T - 58 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.47.ac_acg
53$D_{4}$ \( 1 - 7 T + 80 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.53.ah_dc
59$D_{4}$ \( 1 + 6 T + 110 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.59.g_eg
61$D_{4}$ \( 1 + 21 T + 228 T^{2} + 21 p T^{3} + p^{2} T^{4} \) 2.61.v_iu
67$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.67.a_fe
71$D_{4}$ \( 1 + 5 T + 110 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.71.f_eg
73$D_{4}$ \( 1 + 6 T + 138 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.73.g_fi
79$D_{4}$ \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.79.ac_g
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.83.m_hu
89$D_{4}$ \( 1 - 7 T + 152 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.89.ah_fw
97$D_{4}$ \( 1 + 18 T + 258 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.97.s_jy
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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

−11.18993945231986005536130148749, −11.15935061058093832301240317728, −10.53868572387425541371651565523, −10.16934149193692953348718478314, −9.366448500448303447278781407551, −8.943436852860649154361555461279, −8.385717926981896984627248191318, −8.248511389306582015992965114694, −7.80066345135095209467174389858, −7.53968803805889918278825371013, −6.81167398105403617478743095018, −6.04600020955225623808666992595, −5.71955876290085416171426536336, −4.96565405982927898387085314234, −4.31796503557415797619480641027, −4.27501589463224150581335249703, −3.16810579925613754616026271377, −2.86225181205394860553829947599, −1.80645822875376192796414899877, −1.04117611087261039962427272407, 1.04117611087261039962427272407, 1.80645822875376192796414899877, 2.86225181205394860553829947599, 3.16810579925613754616026271377, 4.27501589463224150581335249703, 4.31796503557415797619480641027, 4.96565405982927898387085314234, 5.71955876290085416171426536336, 6.04600020955225623808666992595, 6.81167398105403617478743095018, 7.53968803805889918278825371013, 7.80066345135095209467174389858, 8.248511389306582015992965114694, 8.385717926981896984627248191318, 8.943436852860649154361555461279, 9.366448500448303447278781407551, 10.16934149193692953348718478314, 10.53868572387425541371651565523, 11.15935061058093832301240317728, 11.18993945231986005536130148749

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