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Properties

Label 4-640332-1.1-c1e2-0-10
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 + 12-s + 16-s + 6·25-s + 27-s + 16·31-s + 36-s − 12·37-s + 48-s − 49-s + 64-s + 8·67-s + 6·75-s + 81-s + 16·93-s − 4·97-s + 6·100-s + 16·103-s + 108-s − 12·111-s − 11·121-s + 16·124-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/2·4-s + 1/3·9-s + 0.288·12-s + 1/4·16-s + 6/5·25-s + 0.192·27-s + 2.87·31-s + 1/6·36-s − 1.97·37-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 + 1.65·93-s − 0.406·97-s + 3/5·100-s + 1.57·103-s + 0.0962·108-s − 1.13·111-s − 121-s + 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-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

\(L(1)\) \(\approx\) \(2.997767198\)
\(L(\frac12)\) \(\approx\) \(2.997767198\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.a_ag
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.13.a_aw
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$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.23.a_bq
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.31.aq_ew
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.37.m_eg
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.43.a_o
47$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.47.a_dm
53$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.53.a_abm
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.a_dy
61$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \) 2.61.a_adi
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.67.ai_fu
71$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.71.a_ec
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.a_eg
79$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \) 2.79.a_afm
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.83.a_fu
89$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.89.a_afq
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.97.e_hq
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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

−8.427320118496747297923118784405, −7.945277664925413146869578375537, −7.54714567060291177997240636945, −6.91860101663622022839962948199, −6.65671387473043484663124717512, −6.30995717749514047642236002575, −5.61118798593894762517062609411, −5.04505323264218432109500685486, −4.66936862862104140298648216317, −4.05846835019594605953583132544, −3.40045925378498278687049642657, −2.93483398088269947909595913869, −2.44212016027147018945820046564, −1.68761492412906607070083117625, −0.878331981291367230472612234111, 0.878331981291367230472612234111, 1.68761492412906607070083117625, 2.44212016027147018945820046564, 2.93483398088269947909595913869, 3.40045925378498278687049642657, 4.05846835019594605953583132544, 4.66936862862104140298648216317, 5.04505323264218432109500685486, 5.61118798593894762517062609411, 6.30995717749514047642236002575, 6.65671387473043484663124717512, 6.91860101663622022839962948199, 7.54714567060291177997240636945, 7.945277664925413146869578375537, 8.427320118496747297923118784405

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