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Properties

Label 4-390e2-1.1-c1e2-0-10
Degree $4$
Conductor $152100$
Sign $-1$
Analytic cond. $9.69802$
Root an. cond. $1.76469$
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 − 8·7-s + 9-s − 2·12-s + 2·13-s + 16-s + 4·19-s + 16·21-s + 25-s + 4·27-s − 8·28-s + 4·31-s + 36-s + 4·37-s − 4·39-s + 4·43-s − 2·48-s + 34·49-s + 2·52-s − 8·57-s + 4·61-s − 8·63-s + 64-s − 8·67-s − 20·73-s − 2·75-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/2·4-s − 3.02·7-s + 1/3·9-s − 0.577·12-s + 0.554·13-s + 1/4·16-s + 0.917·19-s + 3.49·21-s + 1/5·25-s + 0.769·27-s − 1.51·28-s + 0.718·31-s + 1/6·36-s + 0.657·37-s − 0.640·39-s + 0.609·43-s − 0.288·48-s + 34/7·49-s + 0.277·52-s − 1.05·57-s + 0.512·61-s − 1.00·63-s + 1/8·64-s − 0.977·67-s − 2.34·73-s − 0.230·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(152100\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(9.69802\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 152100,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + 2 T + p T^{2} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
13$C_1$ \( ( 1 - T )^{2} \)
good7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.7.i_be
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.11.a_ao
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.19.ae_bq
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.31.ae_co
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.43.ae_dm
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.a_aby
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.59.a_de
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.61.ae_ew
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.67.i_fu
71$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.71.a_ec
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.73.u_jm
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.79.i_gs
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.a_fm
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

−9.151877785951146734411403956564, −8.747160169511579483930127139003, −7.995215963426200914412120114699, −7.13284046687725717765533260434, −7.07291414966298812217331341983, −6.29345621975886340324442998465, −6.21865412034335443989351794245, −5.78605651391475461678969846796, −5.20912025979554566633790456489, −4.28250035270549478393778080408, −3.68007626878963657540665398656, −2.87374848181607439503651528979, −2.83582044678427759968275472285, −1.10216868918993511634847670128, 0, 1.10216868918993511634847670128, 2.83582044678427759968275472285, 2.87374848181607439503651528979, 3.68007626878963657540665398656, 4.28250035270549478393778080408, 5.20912025979554566633790456489, 5.78605651391475461678969846796, 6.21865412034335443989351794245, 6.29345621975886340324442998465, 7.07291414966298812217331341983, 7.13284046687725717765533260434, 7.995215963426200914412120114699, 8.747160169511579483930127139003, 9.151877785951146734411403956564

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