Properties

Label 4-1331200-1.1-c1e2-0-9
Degree $4$
Conductor $1331200$
Sign $1$
Analytic cond. $84.8784$
Root an. cond. $3.03528$
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  − 4·9-s + 5·13-s + 6·17-s − 5·25-s + 6·29-s − 14·49-s + 12·53-s − 2·61-s + 7·81-s + 6·89-s + 24·101-s + 4·109-s + 12·113-s − 20·117-s − 4·121-s + ⋯
L(s)  = 1  − 4/3·9-s + 1.38·13-s + 1.45·17-s − 25-s + 1.11·29-s − 2·49-s + 1.64·53-s − 0.256·61-s + 7/9·81-s + 0.635·89-s + 2.38·101-s + 0.383·109-s + 1.12·113-s − 1.84·117-s − 0.363·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1331200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1331200\)    =    \(2^{12} \cdot 5^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(84.8784\)
Root analytic conductor: \(3.03528\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1331200,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.933424563\)
\(L(\frac12)\) \(\approx\) \(1.933424563\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( 1 + p T^{2} \)
13$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 4 T + p T^{2} ) \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.3.a_e
7$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
11$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.11.a_e
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.17.ag_bi
19$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.19.a_q
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.23.a_ak
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.29.ag_cg
31$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.31.a_aba
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.a_cs
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2^2$ \( 1 - 76 T^{2} + p^{2} T^{4} \) 2.43.a_acy
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.47.a_by
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2^2$ \( 1 - 80 T^{2} + p^{2} T^{4} \) 2.59.a_adc
61$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.c_bq
67$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.67.a_ba
71$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.71.a_ec
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.73.a_fm
79$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.79.a_ao
83$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.83.a_acg
89$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.ag_ec
97$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.97.a_fa
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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

−8.037008401344146816833857600983, −7.63967409663736753519803613458, −7.18648618624298985957887605825, −6.44654619600337466970228073082, −6.19368241809217766423595933116, −5.86410258485463267145856983724, −5.38220612165795310330874235373, −4.97602664093829690279170150084, −4.33179786859627381297729259716, −3.67375187917866482529537773715, −3.35651030575537983560118304136, −2.90900062120407787373056199544, −2.18214146092741720939334261761, −1.42292350550642483822621584682, −0.64245331751713429807999368986, 0.64245331751713429807999368986, 1.42292350550642483822621584682, 2.18214146092741720939334261761, 2.90900062120407787373056199544, 3.35651030575537983560118304136, 3.67375187917866482529537773715, 4.33179786859627381297729259716, 4.97602664093829690279170150084, 5.38220612165795310330874235373, 5.86410258485463267145856983724, 6.19368241809217766423595933116, 6.44654619600337466970228073082, 7.18648618624298985957887605825, 7.63967409663736753519803613458, 8.037008401344146816833857600983

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