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Properties

Label 4-1950e2-1.1-c1e2-0-14
Degree $4$
Conductor $3802500$
Sign $1$
Analytic cond. $242.450$
Root an. cond. $3.94598$
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  − 2·3-s − 4-s + 3·9-s + 2·12-s + 6·13-s + 16-s + 4·17-s − 8·23-s − 4·27-s − 20·29-s − 3·36-s − 12·39-s − 8·43-s − 2·48-s + 10·49-s − 8·51-s − 6·52-s + 12·53-s + 4·61-s − 64-s − 4·68-s + 16·69-s + 5·81-s + 40·87-s + 8·92-s + 4·101-s + 32·103-s + ⋯
L(s)  = 1  − 1.15·3-s − 1/2·4-s + 9-s + 0.577·12-s + 1.66·13-s + 1/4·16-s + 0.970·17-s − 1.66·23-s − 0.769·27-s − 3.71·29-s − 1/2·36-s − 1.92·39-s − 1.21·43-s − 0.288·48-s + 10/7·49-s − 1.12·51-s − 0.832·52-s + 1.64·53-s + 0.512·61-s − 1/8·64-s − 0.485·68-s + 1.92·69-s + 5/9·81-s + 4.28·87-s + 0.834·92-s + 0.398·101-s + 3.15·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3802500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(242.450\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3802500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8554223163\)
\(L(\frac12)\) \(\approx\) \(0.8554223163\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3$C_1$ \( ( 1 + T )^{2} \)
5 \( 1 \)
13$C_2$ \( 1 - 6 T + p T^{2} \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
11$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.11.a_aw
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.17.ae_bm
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.19.a_ac
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.23.i_ck
29$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.29.u_gc
31$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.31.a_bm
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.37.a_ak
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.41.a_s
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.47.a_by
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \) 2.59.a_ady
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.61.ae_ew
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.67.a_afa
71$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.71.a_afm
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.73.a_afa
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \) 2.83.a_afu
89$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \) 2.89.a_afm
97$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.97.a_aby
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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

−9.233566324728389761112545598671, −8.956616267703219010374133862162, −8.872546661394092685504524929219, −7.986726649373977717944633570727, −7.82365178246178017393697956803, −7.58302181044842264236368034285, −6.86151622954446610530335012247, −6.63300166738852675882914303650, −5.95411416925638363048231088412, −5.80448407719749045938899759697, −5.38021803883807541705352987189, −5.28563608774151097138914480545, −4.36036393482824586285885686025, −4.00468394201600375013425422533, −3.67416244534569336265678746531, −3.42392193044851321230971858424, −2.32606893234971786995695680463, −1.76108113140118599472720261700, −1.23703861344529045297577011933, −0.39983921122863822593076142449, 0.39983921122863822593076142449, 1.23703861344529045297577011933, 1.76108113140118599472720261700, 2.32606893234971786995695680463, 3.42392193044851321230971858424, 3.67416244534569336265678746531, 4.00468394201600375013425422533, 4.36036393482824586285885686025, 5.28563608774151097138914480545, 5.38021803883807541705352987189, 5.80448407719749045938899759697, 5.95411416925638363048231088412, 6.63300166738852675882914303650, 6.86151622954446610530335012247, 7.58302181044842264236368034285, 7.82365178246178017393697956803, 7.986726649373977717944633570727, 8.872546661394092685504524929219, 8.956616267703219010374133862162, 9.233566324728389761112545598671

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