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Properties

Label 4-2178e2-1.1-c1e2-0-25
Degree $4$
Conductor $4743684$
Sign $-1$
Analytic cond. $302.461$
Root an. cond. $4.17030$
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  − 3-s − 4-s + 4·5-s + 9-s + 12-s − 4·15-s + 16-s − 4·20-s − 3·23-s + 3·25-s − 27-s + 7·31-s − 36-s + 7·37-s + 4·45-s + 47-s − 48-s − 5·49-s − 2·53-s + 4·59-s + 4·60-s − 64-s − 12·67-s + 3·69-s − 10·71-s − 3·75-s + 4·80-s + ⋯
L(s)  = 1  − 0.577·3-s − 1/2·4-s + 1.78·5-s + 1/3·9-s + 0.288·12-s − 1.03·15-s + 1/4·16-s − 0.894·20-s − 0.625·23-s + 3/5·25-s − 0.192·27-s + 1.25·31-s − 1/6·36-s + 1.15·37-s + 0.596·45-s + 0.145·47-s − 0.144·48-s − 5/7·49-s − 0.274·53-s + 0.520·59-s + 0.516·60-s − 1/8·64-s − 1.46·67-s + 0.361·69-s − 1.18·71-s − 0.346·75-s + 0.447·80-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4743684 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4743684 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4743684\)    =    \(2^{2} \cdot 3^{4} \cdot 11^{4}\)
Sign: $-1$
Analytic conductor: \(302.461\)
Root analytic conductor: \(4.17030\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 4743684,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + T^{2} \)
3$C_1$ \( 1 + T \)
11 \( 1 \)
good5$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.5.ae_n
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.7.a_f
13$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.13.a_q
17$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.17.a_k
19$C_2^2$ \( 1 - 29 T^{2} + p^{2} T^{4} \) 2.19.a_abd
23$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.23.d_bk
29$C_2^2$ \( 1 - 40 T^{2} + p^{2} T^{4} \) 2.29.a_abo
31$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.31.ah_cu
37$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.37.ah_dg
41$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.41.a_ai
43$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \) 2.43.a_l
47$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.47.ab_do
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.53.c_cg
59$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.59.ae_eo
61$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.61.a_cs
67$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.67.m_gk
71$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.71.k_gk
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.a_eg
79$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.79.a_ac
83$C_2^2$ \( 1 + 51 T^{2} + p^{2} T^{4} \) 2.83.a_bz
89$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.89.o_is
97$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.97.ae_du
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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

−7.04504923107661563566253506647, −6.49141536259348474128394771099, −6.19270277946338311660710780227, −6.01451458732580579786497950016, −5.48345282151540046844034431414, −5.25130956195694648950120579494, −4.72589676521015940258052707661, −4.16605860607911809123663176721, −4.01823406321749541924750792853, −3.03304960674054616361116056002, −2.74697347041370747189223301906, −2.07117442686625292175632157778, −1.59027175168058351912024632179, −1.03668606720054659738211398892, 0, 1.03668606720054659738211398892, 1.59027175168058351912024632179, 2.07117442686625292175632157778, 2.74697347041370747189223301906, 3.03304960674054616361116056002, 4.01823406321749541924750792853, 4.16605860607911809123663176721, 4.72589676521015940258052707661, 5.25130956195694648950120579494, 5.48345282151540046844034431414, 6.01451458732580579786497950016, 6.19270277946338311660710780227, 6.49141536259348474128394771099, 7.04504923107661563566253506647

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