Properties

Label 4-3840e2-1.1-c1e2-0-22
Degree $4$
Conductor $14745600$
Sign $1$
Analytic cond. $940.192$
Root an. cond. $5.53737$
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  − 9-s + 4·17-s − 25-s − 20·41-s + 16·47-s − 14·49-s + 16·71-s − 20·73-s + 81-s + 12·89-s + 4·97-s + 32·103-s + 4·113-s + 6·121-s + ⋯
L(s)  = 1  − 1/3·9-s + 0.970·17-s − 1/5·25-s − 3.12·41-s + 2.33·47-s − 2·49-s + 1.89·71-s − 2.34·73-s + 1/9·81-s + 1.27·89-s + 0.406·97-s + 3.15·103-s + 0.376·113-s + 6/11·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(14745600\)    =    \(2^{16} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(940.192\)
Root analytic conductor: \(5.53737\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 14745600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.961688882\)
\(L(\frac12)\) \(\approx\) \(1.961688882\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 + T^{2} \)
good7$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.11.a_ag
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.13.a_aw
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.17.ae_bm
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.19.a_aw
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \) 2.29.a_acc
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.37.a_ba
41$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.41.u_ha
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.43.a_acs
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.47.aq_gc
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.53.a_ag
59$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \) 2.59.a_ady
61$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \) 2.61.a_aeo
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.67.a_k
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.71.aq_hy
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.73.u_jm
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.83.a_aw
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_ig
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.97.ae_hq
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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

−8.551463842332336419378041239689, −8.412833569384716939795128549500, −7.84901833642044963591966113597, −7.69042905905780937935050475279, −7.20355571090021147374732360017, −6.86524450594579777159424397377, −6.41097366005321248085380167058, −6.15173484194683459295153413130, −5.59186205326040074770112823850, −5.39289641792661234315080691248, −4.89790522643415220046098074564, −4.61497332374580121720204879283, −4.00813137038554161732793385360, −3.59684878330961196726182090427, −3.10020302982126717907513491282, −3.00649717045108822405187959438, −1.96457036027455283017002482678, −1.96150747691952472582251646366, −1.10871868552731782992506529266, −0.44274378753751550386415797370, 0.44274378753751550386415797370, 1.10871868552731782992506529266, 1.96150747691952472582251646366, 1.96457036027455283017002482678, 3.00649717045108822405187959438, 3.10020302982126717907513491282, 3.59684878330961196726182090427, 4.00813137038554161732793385360, 4.61497332374580121720204879283, 4.89790522643415220046098074564, 5.39289641792661234315080691248, 5.59186205326040074770112823850, 6.15173484194683459295153413130, 6.41097366005321248085380167058, 6.86524450594579777159424397377, 7.20355571090021147374732360017, 7.69042905905780937935050475279, 7.84901833642044963591966113597, 8.412833569384716939795128549500, 8.551463842332336419378041239689

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