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Properties

Label 4-1110e2-1.1-c1e2-0-24
Degree $4$
Conductor $1232100$
Sign $1$
Analytic cond. $78.5597$
Root an. cond. $2.97714$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s − 5-s − 6-s − 3·7-s + 8-s + 10-s − 4·11-s + 5·13-s + 3·14-s − 15-s − 16-s − 2·17-s − 19-s − 3·21-s + 4·22-s − 4·23-s + 24-s − 5·26-s − 27-s − 10·29-s + 30-s − 8·31-s − 4·33-s + 2·34-s + 3·35-s − 37-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 0.447·5-s − 0.408·6-s − 1.13·7-s + 0.353·8-s + 0.316·10-s − 1.20·11-s + 1.38·13-s + 0.801·14-s − 0.258·15-s − 1/4·16-s − 0.485·17-s − 0.229·19-s − 0.654·21-s + 0.852·22-s − 0.834·23-s + 0.204·24-s − 0.980·26-s − 0.192·27-s − 1.85·29-s + 0.182·30-s − 1.43·31-s − 0.696·33-s + 0.342·34-s + 0.507·35-s − 0.164·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1232100\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(78.5597\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1232100,\ (\ :1/2, 1/2),\ 1)\)

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$C_2$ \( 1 + T + T^{2} \)
3$C_2$ \( 1 - T + T^{2} \)
5$C_2$ \( 1 + T + T^{2} \)
37$C_2$ \( 1 + T + p T^{2} \)
good7$C_2^2$ \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.7.d_c
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.11.e_ba
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.af_m
17$C_2^2$ \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.17.c_an
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.19.b_as
23$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.23.e_by
29$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.29.k_df
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.31.i_da
41$C_2^2$ \( 1 + 10 T + 59 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.41.k_ch
43$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.43.m_es
47$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.47.y_je
53$C_2^2$ \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.53.g_ar
59$C_2^2$ \( 1 + 10 T + 41 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.59.k_bp
61$C_2^2$ \( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.61.ae_abt
67$C_2^2$ \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.67.c_acl
71$C_2^2$ \( 1 - T - 70 T^{2} - p T^{3} + p^{2} T^{4} \) 2.71.ab_acs
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.73.am_ha
79$C_2^2$ \( 1 + 10 T + 21 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.79.k_v
83$C_2^2$ \( 1 - 13 T + 86 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.83.an_di
89$C_2^2$ \( 1 + 16 T + 167 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.89.q_gl
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.97.au_li
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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

−9.469431670114961762918672593103, −9.320053072555493080117409120304, −8.824643971760549332358762440473, −8.277500539204634583840400575198, −8.117795326298462961192202934998, −7.88481517239625029123951184546, −7.12310953446848890489444829599, −6.89180257601775825436961282401, −6.28913424501088577067100030500, −6.01780284818166202160259939497, −5.22761787903985956667123477545, −5.05068949336486866216640329989, −4.20002343178938403091050157208, −3.58224726861927794961352993671, −3.43510292531433566086146437629, −2.95283374525383741297836899869, −1.80379910684588143234779557876, −1.80206367839233783961982228385, 0, 0, 1.80206367839233783961982228385, 1.80379910684588143234779557876, 2.95283374525383741297836899869, 3.43510292531433566086146437629, 3.58224726861927794961352993671, 4.20002343178938403091050157208, 5.05068949336486866216640329989, 5.22761787903985956667123477545, 6.01780284818166202160259939497, 6.28913424501088577067100030500, 6.89180257601775825436961282401, 7.12310953446848890489444829599, 7.88481517239625029123951184546, 8.117795326298462961192202934998, 8.277500539204634583840400575198, 8.824643971760549332358762440473, 9.320053072555493080117409120304, 9.469431670114961762918672593103

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