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Properties

Label 4-1197900-1.1-c1e2-0-28
Degree $4$
Conductor $1197900$
Sign $-1$
Analytic cond. $76.3791$
Root an. cond. $2.95626$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 4-s + 2·5-s + 3·9-s + 11-s − 2·12-s − 4·15-s + 16-s + 2·20-s + 3·25-s − 4·27-s − 16·31-s − 2·33-s + 3·36-s − 20·37-s + 44-s + 6·45-s − 2·48-s + 2·49-s + 28·53-s + 2·55-s − 8·59-s − 4·60-s + 64-s − 8·67-s + 16·71-s − 6·75-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/2·4-s + 0.894·5-s + 9-s + 0.301·11-s − 0.577·12-s − 1.03·15-s + 1/4·16-s + 0.447·20-s + 3/5·25-s − 0.769·27-s − 2.87·31-s − 0.348·33-s + 1/2·36-s − 3.28·37-s + 0.150·44-s + 0.894·45-s − 0.288·48-s + 2/7·49-s + 3.84·53-s + 0.269·55-s − 1.04·59-s − 0.516·60-s + 1/8·64-s − 0.977·67-s + 1.89·71-s − 0.692·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1197900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1197900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1197900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 11^{3}\)
Sign: $-1$
Analytic conductor: \(76.3791\)
Root analytic conductor: \(2.95626\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1197900,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_1$ \( ( 1 + T )^{2} \)
5$C_1$ \( ( 1 - T )^{2} \)
11$C_1$ \( 1 - T \)
good7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.a_w
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.a_be
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.19.a_aba
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.29.a_cc
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.31.q_ew
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.37.u_gs
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.a_as
43$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.43.a_di
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.53.abc_lq
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.59.i_fe
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.61.a_acw
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.67.i_fu
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.71.aq_hy
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.a_bu
79$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.79.a_o
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.83.a_fu
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.89.m_ig
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.97.bc_pa
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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

−7.40696925033699638160691249365, −7.27752953900878357828554051261, −6.87984264875203216093880454454, −6.59167185110691005638218150143, −5.88014804790306627418533622087, −5.55307889866064846829275639149, −5.39233288337080961598018021053, −4.92089940314369655781504062688, −4.06246632062950272760150417569, −3.75484499233248573080193197253, −3.12548788757864162693227054094, −2.18827740998320422124385016029, −1.85410703674039090636159152963, −1.17775005180092691264201056199, 0, 1.17775005180092691264201056199, 1.85410703674039090636159152963, 2.18827740998320422124385016029, 3.12548788757864162693227054094, 3.75484499233248573080193197253, 4.06246632062950272760150417569, 4.92089940314369655781504062688, 5.39233288337080961598018021053, 5.55307889866064846829275639149, 5.88014804790306627418533622087, 6.59167185110691005638218150143, 6.87984264875203216093880454454, 7.27752953900878357828554051261, 7.40696925033699638160691249365

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