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Properties

Label 4-48e3-1.1-c1e2-0-5
Degree $4$
Conductor $110592$
Sign $1$
Analytic cond. $7.05144$
Root an. cond. $1.62955$
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 + 4·7-s + 9-s + 4·13-s − 4·21-s + 2·25-s − 27-s + 4·31-s + 12·37-s − 4·39-s − 16·43-s + 2·49-s + 12·61-s + 4·63-s + 8·67-s − 12·73-s − 2·75-s − 12·79-s + 81-s + 16·91-s − 4·93-s + 20·97-s − 4·103-s − 12·109-s − 12·111-s + 4·117-s + 10·121-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.51·7-s + 1/3·9-s + 1.10·13-s − 0.872·21-s + 2/5·25-s − 0.192·27-s + 0.718·31-s + 1.97·37-s − 0.640·39-s − 2.43·43-s + 2/7·49-s + 1.53·61-s + 0.503·63-s + 0.977·67-s − 1.40·73-s − 0.230·75-s − 1.35·79-s + 1/9·81-s + 1.67·91-s − 0.414·93-s + 2.03·97-s − 0.394·103-s − 1.14·109-s − 1.13·111-s + 0.369·117-s + 0.909·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 110592 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110592 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(110592\)    =    \(2^{12} \cdot 3^{3}\)
Sign: $1$
Analytic conductor: \(7.05144\)
Root analytic conductor: \(1.62955\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 110592,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.661003709\)
\(L(\frac12)\) \(\approx\) \(1.661003709\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$ \( 1 + T \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
7$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.7.ae_o
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.11.a_ak
13$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.ae_o
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.17.a_as
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.23.a_abi
29$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.29.a_abi
31$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.31.ae_ck
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.37.am_dq
41$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.41.a_ck
43$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.43.q_fe
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.a_aby
53$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \) 2.53.a_da
59$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.59.a_bm
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.61.am_fm
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.67.ai_fu
71$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \) 2.71.a_aco
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.73.m_ha
79$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.m_hi
83$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.83.a_acg
89$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.89.a_as
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.97.au_li
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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

−9.646683558532825422961594747295, −8.749141058887360976444729349389, −8.584225029305632486131619689414, −7.979884906529064995238642161307, −7.70829270802114696023380247265, −6.89921642544177574384285962302, −6.46995102026997323412602058486, −5.94034673079163869136640153924, −5.30063304206017561416271301535, −4.85019083747205177959207048181, −4.34350790247382349556382255708, −3.69366492145467467963103303891, −2.80096801481351231668711235893, −1.80088478355185876366952450636, −1.08316865159020613155446044444, 1.08316865159020613155446044444, 1.80088478355185876366952450636, 2.80096801481351231668711235893, 3.69366492145467467963103303891, 4.34350790247382349556382255708, 4.85019083747205177959207048181, 5.30063304206017561416271301535, 5.94034673079163869136640153924, 6.46995102026997323412602058486, 6.89921642544177574384285962302, 7.70829270802114696023380247265, 7.979884906529064995238642161307, 8.584225029305632486131619689414, 8.749141058887360976444729349389, 9.646683558532825422961594747295

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