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Properties

Label 4-8575-1.1-c1e2-0-2
Degree $4$
Conductor $8575$
Sign $-1$
Analytic cond. $0.546749$
Root an. cond. $0.859898$
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  − 4·4-s + 7-s − 5·9-s − 6·11-s + 12·16-s − 12·23-s + 25-s − 4·28-s + 6·29-s + 20·36-s + 4·37-s − 20·43-s + 24·44-s + 49-s + 24·53-s − 5·63-s − 32·64-s − 8·67-s − 6·77-s − 2·79-s + 16·81-s + 48·92-s + 30·99-s − 4·100-s + 12·107-s − 14·109-s + 12·112-s + ⋯
L(s)  = 1  − 2·4-s + 0.377·7-s − 5/3·9-s − 1.80·11-s + 3·16-s − 2.50·23-s + 1/5·25-s − 0.755·28-s + 1.11·29-s + 10/3·36-s + 0.657·37-s − 3.04·43-s + 3.61·44-s + 1/7·49-s + 3.29·53-s − 0.629·63-s − 4·64-s − 0.977·67-s − 0.683·77-s − 0.225·79-s + 16/9·81-s + 5.00·92-s + 3.01·99-s − 2/5·100-s + 1.16·107-s − 1.34·109-s + 1.13·112-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8575 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8575 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8575\)    =    \(5^{2} \cdot 7^{3}\)
Sign: $-1$
Analytic conductor: \(0.546749\)
Root analytic conductor: \(0.859898\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 8575,\ (\ :1/2, 1/2),\ -1)\)

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 $\mathbf{F}_p$
bad5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_1$ \( 1 - T \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.2.a_e
3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.3.a_f
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.11.g_bf
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.13.a_b
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.a_z
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.19.a_bi
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.23.m_de
29$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.29.ag_cp
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.41.a_ack
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.43.u_he
47$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.47.a_n
53$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.53.ay_jq
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.61.a_cg
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.67.i_fu
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.73.a_fm
79$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.79.c_gd
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.a_w
89$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.89.a_bi
97$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.97.a_hl
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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

−11.72263911909931664995468447198, −10.46766423688804548750268495569, −10.26175613408033680908306725927, −9.833804192038460581265801340645, −8.722193961422815471879095064652, −8.679498853252053444462430241617, −7.980947296320422939728903060714, −7.80156257853413862136981277007, −6.30078072165756181986797640838, −5.47907564040983295563221596641, −5.30121167828360660438204787375, −4.48390865811726709341858080699, −3.61689003045514298236484607491, −2.61198254943438177491615875407, 0, 2.61198254943438177491615875407, 3.61689003045514298236484607491, 4.48390865811726709341858080699, 5.30121167828360660438204787375, 5.47907564040983295563221596641, 6.30078072165756181986797640838, 7.80156257853413862136981277007, 7.980947296320422939728903060714, 8.679498853252053444462430241617, 8.722193961422815471879095064652, 9.833804192038460581265801340645, 10.26175613408033680908306725927, 10.46766423688804548750268495569, 11.72263911909931664995468447198

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