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Properties

Label 4-47104-1.1-c1e2-0-11
Degree $4$
Conductor $47104$
Sign $-1$
Analytic cond. $3.00339$
Root an. cond. $1.31644$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·7-s − 9-s − 3·17-s − 3·23-s + 3·25-s − 9·31-s − 3·41-s − 15·47-s − 49-s + 4·63-s + 6·71-s + 10·73-s + 11·79-s − 8·81-s + 89-s − 14·97-s − 14·103-s − 5·113-s + 12·119-s − 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 3·153-s + ⋯
L(s)  = 1  − 1.51·7-s − 1/3·9-s − 0.727·17-s − 0.625·23-s + 3/5·25-s − 1.61·31-s − 0.468·41-s − 2.18·47-s − 1/7·49-s + 0.503·63-s + 0.712·71-s + 1.17·73-s + 1.23·79-s − 8/9·81-s + 0.105·89-s − 1.42·97-s − 1.37·103-s − 0.470·113-s + 1.10·119-s − 0.727·121-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 + 0.242·153-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 47104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47104 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(47104\)    =    \(2^{11} \cdot 23\)
Sign: $-1$
Analytic conductor: \(3.00339\)
Root analytic conductor: \(1.31644\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 47104,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
23$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 2 T + p T^{2} ) \)
good3$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.3.a_b
5$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \) 2.5.a_ad
7$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.7.e_r
11$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.11.a_i
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.d_bi
19$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.19.a_au
29$C_2^2$ \( 1 + 56 T^{2} + p^{2} T^{4} \) 2.29.a_ce
31$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.31.j_bo
37$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \) 2.37.a_acc
41$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.d_cm
43$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.43.a_au
47$C_2$$\times$$C_2$ \( ( 1 + 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.p_fu
53$C_2^2$ \( 1 + 37 T^{2} + p^{2} T^{4} \) 2.53.a_bl
59$C_2^2$ \( 1 - 73 T^{2} + p^{2} T^{4} \) 2.59.a_acv
61$C_2^2$ \( 1 - 57 T^{2} + p^{2} T^{4} \) 2.61.a_acf
67$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.67.a_aby
71$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.71.ag_fm
73$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.73.ak_go
79$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + p T^{2} ) \) 2.79.al_gc
83$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.83.a_au
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.89.ab_dk
97$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.97.o_ik
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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

−9.827770512483804718651225539447, −9.311910481094263326170762000842, −9.127764818643200407228133753775, −8.221147607057683778288147093195, −8.006804510768835137402978325359, −7.01213076363519041587621672138, −6.69011198418340330912888544255, −6.29694145347950111842640025364, −5.53881289013821234015893341966, −5.00489182792217975180365964670, −4.10231098648888242554554109927, −3.45114983003963330872254335710, −2.90034422115385905871751020075, −1.87633009609720075442416851463, 0, 1.87633009609720075442416851463, 2.90034422115385905871751020075, 3.45114983003963330872254335710, 4.10231098648888242554554109927, 5.00489182792217975180365964670, 5.53881289013821234015893341966, 6.29694145347950111842640025364, 6.69011198418340330912888544255, 7.01213076363519041587621672138, 8.006804510768835137402978325359, 8.221147607057683778288147093195, 9.127764818643200407228133753775, 9.311910481094263326170762000842, 9.827770512483804718651225539447

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