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Properties

Label 4-2178e2-1.1-c1e2-0-2
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + 2·5-s + 16-s − 2·20-s − 8·23-s − 6·25-s − 4·31-s − 16·37-s + 12·47-s − 4·49-s − 2·53-s + 8·59-s − 64-s + 12·67-s − 4·71-s + 2·80-s − 14·89-s + 8·92-s − 20·97-s + 6·100-s + 8·103-s − 10·113-s − 16·115-s + 4·124-s − 22·125-s + 127-s + 131-s + ⋯
L(s)  = 1  − 1/2·4-s + 0.894·5-s + 1/4·16-s − 0.447·20-s − 1.66·23-s − 6/5·25-s − 0.718·31-s − 2.63·37-s + 1.75·47-s − 4/7·49-s − 0.274·53-s + 1.04·59-s − 1/8·64-s + 1.46·67-s − 0.474·71-s + 0.223·80-s − 1.48·89-s + 0.834·92-s − 2.03·97-s + 3/5·100-s + 0.788·103-s − 0.940·113-s − 1.49·115-s + 0.359·124-s − 1.96·125-s + 0.0887·127-s + 0.0873·131-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: \(0\)
Selberg data: \((4,\ 4743684,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.183114763\)
\(L(\frac12)\) \(\approx\) \(1.183114763\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 \)
good5$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) 2.5.ac_k
7$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.7.a_e
13$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.13.a_i
17$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.17.a_ao
19$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.19.a_ae
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.23.i_ck
29$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.29.a_bm
31$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.31.e_be
37$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.37.q_fi
41$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.41.a_ac
43$C_2^2$ \( 1 - 68 T^{2} + p^{2} T^{4} \) 2.43.a_acq
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.47.am_ew
53$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.c_de
59$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) 2.59.ai_eo
61$C_2^2$ \( 1 - 64 T^{2} + p^{2} T^{4} \) 2.61.a_acm
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.67.am_gk
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.71.e_aby
73$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \) 2.73.a_bo
79$C_2^2$ \( 1 + 68 T^{2} + p^{2} T^{4} \) 2.79.a_cq
83$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.83.a_dy
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$ \( ( 1 + 2 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.97.u_iw
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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

−7.25880401193670853968949983042, −6.96421707904922046460959527427, −6.50874095262334639204969871055, −5.97304185807736089718201740444, −5.67279459702140560643133665203, −5.39526573017630313768557396652, −5.01952518168687158823604387423, −4.29739132372177016922105444504, −3.90092568754038926515015500394, −3.70018394445048847314239593179, −2.97610663533230995764882213018, −2.32131094825811409526589663339, −1.88046497446053683914396307779, −1.48669786726241911306163497192, −0.36908303254872293574197186979, 0.36908303254872293574197186979, 1.48669786726241911306163497192, 1.88046497446053683914396307779, 2.32131094825811409526589663339, 2.97610663533230995764882213018, 3.70018394445048847314239593179, 3.90092568754038926515015500394, 4.29739132372177016922105444504, 5.01952518168687158823604387423, 5.39526573017630313768557396652, 5.67279459702140560643133665203, 5.97304185807736089718201740444, 6.50874095262334639204969871055, 6.96421707904922046460959527427, 7.25880401193670853968949983042

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