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Properties

Label 4-59150-1.1-c1e2-0-1
Degree $4$
Conductor $59150$
Sign $1$
Analytic cond. $3.77145$
Root an. cond. $1.39356$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + 2·5-s + 5·7-s + 2·8-s + 2·9-s − 2·13-s + 16-s − 2·20-s − 25-s − 5·28-s − 4·32-s + 10·35-s − 2·36-s + 4·37-s + 4·40-s + 4·45-s − 20·47-s + 14·49-s + 2·52-s + 10·56-s + 4·61-s + 10·63-s + 3·64-s − 4·65-s − 28·67-s + 4·72-s − 4·73-s + ⋯
L(s)  = 1  − 1/2·4-s + 0.894·5-s + 1.88·7-s + 0.707·8-s + 2/3·9-s − 0.554·13-s + 1/4·16-s − 0.447·20-s − 1/5·25-s − 0.944·28-s − 0.707·32-s + 1.69·35-s − 1/3·36-s + 0.657·37-s + 0.632·40-s + 0.596·45-s − 2.91·47-s + 2·49-s + 0.277·52-s + 1.33·56-s + 0.512·61-s + 1.25·63-s + 3/8·64-s − 0.496·65-s − 3.42·67-s + 0.471·72-s − 0.468·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 59150 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59150 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(59150\)    =    \(2 \cdot 5^{2} \cdot 7 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(3.77145\)
Root analytic conductor: \(1.39356\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 59150,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.954895180\)
\(L(\frac12)\) \(\approx\) \(1.954895180\)
\(L(\frac{3}{2})\) 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_2$ \( ( 1 - T )( 1 + T + p T^{2} ) \)
5$C_2$ \( 1 - 2 T + p T^{2} \)
7$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 4 T + p T^{2} ) \)
13$C_2$ \( 1 + 2 T + p T^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.3.a_ac
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.11.a_ao
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.a_be
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.19.a_ac
23$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.23.a_ac
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.29.a_cc
31$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.31.a_aw
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.ae_ck
41$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.41.a_ac
43$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.43.a_by
47$C_2$$\times$$C_2$ \( ( 1 + 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.u_hi
53$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.53.a_o
59$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.59.a_ck
61$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.61.ae_eg
67$C_2$$\times$$C_2$ \( ( 1 + 12 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.67.bc_mo
71$C_2^2$ \( 1 + 118 T^{2} + p^{2} T^{4} \) 2.71.a_eo
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.73.e_g
79$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.m_hi
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.a_w
89$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.89.a_ck
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.97.i_gs
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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

−10.01680481248676236476980323249, −9.605275744492367615540507161613, −8.875484026397415534891822107632, −8.481424311947461205547125578809, −7.82369930024139171738123114373, −7.58068525353008675932779773298, −6.97905507864066321638177931695, −6.18794458648678176742830536399, −5.53295912747360394652127388609, −5.01084077846916460603868436209, −4.53694860945779743314912752035, −4.16597088998770786434292076864, −2.96277527305276669461039847338, −1.80857614108946620467904953688, −1.52874685361915009751032215020, 1.52874685361915009751032215020, 1.80857614108946620467904953688, 2.96277527305276669461039847338, 4.16597088998770786434292076864, 4.53694860945779743314912752035, 5.01084077846916460603868436209, 5.53295912747360394652127388609, 6.18794458648678176742830536399, 6.97905507864066321638177931695, 7.58068525353008675932779773298, 7.82369930024139171738123114373, 8.481424311947461205547125578809, 8.875484026397415534891822107632, 9.605275744492367615540507161613, 10.01680481248676236476980323249

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