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Properties

Label 4-640332-1.1-c1e2-0-5
Degree $4$
Conductor $640332$
Sign $1$
Analytic cond. $40.8281$
Root an. cond. $2.52778$
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  + 3-s + 4-s + 9-s − 6·11-s + 12-s + 16-s + 6·25-s + 27-s + 6·29-s − 2·31-s − 6·33-s + 36-s + 6·37-s − 6·41-s − 6·44-s + 48-s − 49-s + 64-s + 8·67-s + 6·75-s + 81-s + 6·87-s − 2·93-s − 4·97-s − 6·99-s + 6·100-s + 12·101-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/2·4-s + 1/3·9-s − 1.80·11-s + 0.288·12-s + 1/4·16-s + 6/5·25-s + 0.192·27-s + 1.11·29-s − 0.359·31-s − 1.04·33-s + 1/6·36-s + 0.986·37-s − 0.937·41-s − 0.904·44-s + 0.144·48-s − 1/7·49-s + 1/8·64-s + 0.977·67-s + 0.692·75-s + 1/9·81-s + 0.643·87-s − 0.207·93-s − 0.406·97-s − 0.603·99-s + 3/5·100-s + 1.19·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640332 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640332 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(640332\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(40.8281\)
Root analytic conductor: \(2.52778\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 640332,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.422568772\)
\(L(\frac12)\) \(\approx\) \(2.422568772\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
7$C_2$ \( 1 + T^{2} \)
11$C_2$ \( 1 + 6 T + p T^{2} \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.a_ag
13$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.13.a_ae
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.a_be
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.19.a_bi
23$C_2^2$ \( 1 + 24 T^{2} + p^{2} T^{4} \) 2.23.a_y
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.29.ag_cg
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.31.c_as
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.37.ag_cw
41$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.g_de
43$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.43.a_o
47$C_2^2$ \( 1 - 36 T^{2} + p^{2} T^{4} \) 2.47.a_abk
53$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \) 2.53.a_acw
59$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.59.a_ag
61$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \) 2.61.a_bo
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.67.ai_fu
71$C_2^2$ \( 1 + 124 T^{2} + p^{2} T^{4} \) 2.71.a_eu
73$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.73.a_c
79$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.79.a_bm
83$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.83.a_abe
89$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \) 2.89.a_abm
97$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.e_cc
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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

−8.295610574610407759725407849958, −7.944001850443576619202935956012, −7.54933157318806943127813292354, −7.11786920602932915026426529419, −6.59200975973742239022958505327, −6.24269726665450571638474838643, −5.48015905701788430969278780413, −5.18520795527895819484772342341, −4.66494399589857604857345682734, −4.12644275773749040094829363026, −3.25181356993824591591533447086, −2.97011757475229408648459275885, −2.43371316457661695605742618325, −1.81765070010737079559308864452, −0.74575401388116724502101325022, 0.74575401388116724502101325022, 1.81765070010737079559308864452, 2.43371316457661695605742618325, 2.97011757475229408648459275885, 3.25181356993824591591533447086, 4.12644275773749040094829363026, 4.66494399589857604857345682734, 5.18520795527895819484772342341, 5.48015905701788430969278780413, 6.24269726665450571638474838643, 6.59200975973742239022958505327, 7.11786920602932915026426529419, 7.54933157318806943127813292354, 7.944001850443576619202935956012, 8.295610574610407759725407849958

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