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Properties

Label 4-640332-1.1-c1e2-0-8
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 − 10·25-s + 27-s + 8·31-s + 36-s + 4·37-s + 48-s − 49-s + 64-s + 24·67-s − 10·75-s + 81-s + 8·93-s + 12·97-s − 10·100-s + 8·103-s + 108-s + 4·111-s − 11·121-s + 8·124-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 − 2·25-s + 0.192·27-s + 1.43·31-s + 1/6·36-s + 0.657·37-s + 0.144·48-s − 1/7·49-s + 1/8·64-s + 2.93·67-s − 1.15·75-s + 1/9·81-s + 0.829·93-s + 1.21·97-s − 100-s + 0.788·103-s + 0.0962·108-s + 0.379·111-s − 121-s + 0.718·124-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

\(L(1)\) \(\approx\) \(2.751608890\)
\(L(\frac12)\) \(\approx\) \(2.751608890\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
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$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.19.a_abi
23$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.23.a_bq
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.29.a_abq
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.31.ai_da
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.41.a_da
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.43.a_aby
47$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.47.a_ag
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$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \) 2.61.a_aeo
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.67.ay_ks
71$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.71.a_fi
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.73.a_afa
79$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \) 2.79.a_afm
83$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.83.a_dy
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.a_fm
97$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.97.am_iw
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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

−8.165678724980624284870488143424, −7.981890914755466295548007347155, −7.64538706777287483750863078199, −7.00445772168504758614495384146, −6.59548489878523284918699906331, −6.20320870755204265643405675642, −5.65395983582694389889196901818, −5.19773949443833088629459235218, −4.48080129460261086295385087598, −4.07187884298526393281892225726, −3.50422169579393696238500782132, −2.94162149280893570398308614899, −2.27080885217504090145569675475, −1.85015606075908436048505984956, −0.809374100474558800250065491858, 0.809374100474558800250065491858, 1.85015606075908436048505984956, 2.27080885217504090145569675475, 2.94162149280893570398308614899, 3.50422169579393696238500782132, 4.07187884298526393281892225726, 4.48080129460261086295385087598, 5.19773949443833088629459235218, 5.65395983582694389889196901818, 6.20320870755204265643405675642, 6.59548489878523284918699906331, 7.00445772168504758614495384146, 7.64538706777287483750863078199, 7.981890914755466295548007347155, 8.165678724980624284870488143424

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