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Properties

Label 4-2376e2-1.1-c1e2-0-11
Degree $4$
Conductor $5645376$
Sign $1$
Analytic cond. $359.954$
Root an. cond. $4.35573$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·11-s + 4·17-s + 6·23-s − 10·25-s + 8·29-s + 8·31-s − 6·37-s + 4·41-s + 4·43-s + 2·47-s − 7·49-s + 8·53-s + 14·59-s + 16·67-s + 8·71-s − 8·73-s − 8·79-s + 16·83-s + 16·89-s − 6·97-s + 16·101-s + 8·103-s − 4·109-s + 16·113-s + 3·121-s + 127-s + 131-s + ⋯
L(s)  = 1  + 0.603·11-s + 0.970·17-s + 1.25·23-s − 2·25-s + 1.48·29-s + 1.43·31-s − 0.986·37-s + 0.624·41-s + 0.609·43-s + 0.291·47-s − 49-s + 1.09·53-s + 1.82·59-s + 1.95·67-s + 0.949·71-s − 0.936·73-s − 0.900·79-s + 1.75·83-s + 1.69·89-s − 0.609·97-s + 1.59·101-s + 0.788·103-s − 0.383·109-s + 1.50·113-s + 3/11·121-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5645376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5645376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5645376\)    =    \(2^{6} \cdot 3^{6} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(359.954\)
Root analytic conductor: \(4.35573\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5645376,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.275664494\)
\(L(\frac12)\) \(\approx\) \(3.275664494\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
7$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.7.a_h
13$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.13.a_ba
17$D_{4}$ \( 1 - 4 T + 31 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.17.ae_bf
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.19.a_k
23$D_{4}$ \( 1 - 6 T + 27 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_bb
29$D_{4}$ \( 1 - 8 T + 67 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.29.ai_cp
31$D_{4}$ \( 1 - 8 T + 50 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.31.ai_by
37$D_{4}$ \( 1 + 6 T + 55 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.37.g_cd
41$D_{4}$ \( 1 - 4 T + 23 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.41.ae_x
43$D_{4}$ \( 1 - 4 T + 83 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.43.ae_df
47$D_{4}$ \( 1 - 2 T + 67 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.47.ac_cp
53$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.53.ai_es
59$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.59.ao_gl
61$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.61.a_k
67$D_{4}$ \( 1 - 16 T + 170 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.67.aq_go
71$D_{4}$ \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.71.ai_bu
73$D_{4}$ \( 1 + 8 T + 50 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.73.i_by
79$D_{4}$ \( 1 + 8 T + 111 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.79.i_eh
83$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.83.aq_iw
89$D_{4}$ \( 1 - 16 T + 214 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.89.aq_ig
97$D_{4}$ \( 1 + 6 T + 91 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.97.g_dn
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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

−9.130956984783796092837855683055, −8.737466710033693923419995376997, −8.338866775738072023265820327730, −8.169557001691893668595202614151, −7.46648597125565936895465311499, −7.42954494635361612397177094789, −6.72742978745653294905788751257, −6.57529305926087599984769991447, −5.98266915136523680583596954079, −5.77355642766819072496443723062, −5.06027324703766708454231402794, −4.98611360820783035838874059560, −4.23765562054035146935362117736, −3.99887896433681548930191239394, −3.26151716919189775067057285068, −3.20798734261277440837009024484, −2.24932541508482934024250720113, −2.07150697892189722633037777039, −1.00166572906252625647384587322, −0.78734255291912420635360147050, 0.78734255291912420635360147050, 1.00166572906252625647384587322, 2.07150697892189722633037777039, 2.24932541508482934024250720113, 3.20798734261277440837009024484, 3.26151716919189775067057285068, 3.99887896433681548930191239394, 4.23765562054035146935362117736, 4.98611360820783035838874059560, 5.06027324703766708454231402794, 5.77355642766819072496443723062, 5.98266915136523680583596954079, 6.57529305926087599984769991447, 6.72742978745653294905788751257, 7.42954494635361612397177094789, 7.46648597125565936895465311499, 8.169557001691893668595202614151, 8.338866775738072023265820327730, 8.737466710033693923419995376997, 9.130956984783796092837855683055

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