Properties

Label 4-56448-1.1-c1e2-0-6
Degree $4$
Conductor $56448$
Sign $1$
Analytic cond. $3.59917$
Root an. cond. $1.37737$
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  + 9-s + 10·25-s − 4·29-s + 12·37-s − 7·49-s + 12·53-s + 81-s + 28·109-s + 4·113-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  + 1/3·9-s + 2·25-s − 0.742·29-s + 1.97·37-s − 49-s + 1.64·53-s + 1/9·81-s + 2.68·109-s + 0.376·113-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56448 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56448 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(56448\)    =    \(2^{7} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(3.59917\)
Root analytic conductor: \(1.37737\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{56448} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 56448,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.450288378\)
\(L(\frac12)\) \(\approx\) \(1.450288378\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_2$ \( 1 + p T^{2} \)
good5$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2^2$ \( 1 - 162 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
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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

−10.05657115529039899400899719864, −9.469661957727810483808554510835, −8.982762462636603910619370970962, −8.579339151050513374259785543537, −7.87492835830535776060315308327, −7.46665754522012491876914070081, −6.85270032623510979824760622364, −6.38372108486695177916356868283, −5.74169639016154741682757784729, −5.06169873478418323890131448492, −4.53514935876397205520234181056, −3.84950349006191993033927924571, −3.04929903053313537460971819934, −2.30019117285336016957893454817, −1.09032516852600613173339041414, 1.09032516852600613173339041414, 2.30019117285336016957893454817, 3.04929903053313537460971819934, 3.84950349006191993033927924571, 4.53514935876397205520234181056, 5.06169873478418323890131448492, 5.74169639016154741682757784729, 6.38372108486695177916356868283, 6.85270032623510979824760622364, 7.46665754522012491876914070081, 7.87492835830535776060315308327, 8.579339151050513374259785543537, 8.982762462636603910619370970962, 9.469661957727810483808554510835, 10.05657115529039899400899719864

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