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Properties

Label 4-188604-1.1-c1e2-0-5
Degree $4$
Conductor $188604$
Sign $-1$
Analytic cond. $12.0255$
Root an. cond. $1.86219$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 4-s + 2·7-s − 2·9-s − 12-s + 16-s − 14·19-s − 2·21-s + 9·25-s + 5·27-s + 2·28-s − 31-s − 2·36-s − 6·37-s − 14·43-s − 48-s − 7·49-s + 14·57-s − 14·61-s − 4·63-s + 64-s + 8·67-s − 9·75-s − 14·76-s − 10·79-s + 81-s − 2·84-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/2·4-s + 0.755·7-s − 2/3·9-s − 0.288·12-s + 1/4·16-s − 3.21·19-s − 0.436·21-s + 9/5·25-s + 0.962·27-s + 0.377·28-s − 0.179·31-s − 1/3·36-s − 0.986·37-s − 2.13·43-s − 0.144·48-s − 49-s + 1.85·57-s − 1.79·61-s − 0.503·63-s + 1/8·64-s + 0.977·67-s − 1.03·75-s − 1.60·76-s − 1.12·79-s + 1/9·81-s − 0.218·84-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 188604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 188604 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(188604\)    =    \(2^{2} \cdot 3^{2} \cdot 13^{2} \cdot 31\)
Sign: $-1$
Analytic conductor: \(12.0255\)
Root analytic conductor: \(1.86219\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 188604,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + T + p T^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
31$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
good5$C_2^2$ \( 1 - 9 T^{2} + p^{2} T^{4} \) 2.5.a_aj
7$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.7.ac_l
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.a_g
17$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.17.a_az
19$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.19.o_di
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.23.a_s
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.29.a_aw
37$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.37.g_t
41$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.41.a_ac
43$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.43.o_ep
47$C_2^2$ \( 1 - 71 T^{2} + p^{2} T^{4} \) 2.47.a_act
53$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.53.a_ak
59$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \) 2.59.a_acc
61$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.o_gg
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) 2.67.ai_fe
71$C_2^2$ \( 1 - 15 T^{2} + p^{2} T^{4} \) 2.71.a_ap
73$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.73.a_de
79$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.k_gs
83$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.83.a_bu
89$C_2^2$ \( 1 - 110 T^{2} + p^{2} T^{4} \) 2.89.a_aeg
97$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.97.as_kg
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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

−8.725510142251702007655157597224, −8.386564046519987094869650823058, −8.183464464091161694473012129650, −7.36924302044750264200537642515, −6.73916835362946300100258155153, −6.48210999751061385445664933390, −6.11042513925964807210444980598, −5.35727898436416452538116194055, −4.80511740211588002842587080563, −4.56073222219958799038677424172, −3.65003678902768550492447880003, −2.95178514421884618868301690676, −2.20094478945720220195534275788, −1.53806119121660542428288143919, 0, 1.53806119121660542428288143919, 2.20094478945720220195534275788, 2.95178514421884618868301690676, 3.65003678902768550492447880003, 4.56073222219958799038677424172, 4.80511740211588002842587080563, 5.35727898436416452538116194055, 6.11042513925964807210444980598, 6.48210999751061385445664933390, 6.73916835362946300100258155153, 7.36924302044750264200537642515, 8.183464464091161694473012129650, 8.386564046519987094869650823058, 8.725510142251702007655157597224

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