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Properties

Label 4-2376e2-1.1-c1e2-0-3
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·5-s + 4·7-s − 11-s − 6·13-s + 8·17-s + 4·19-s − 4·23-s + 5·25-s + 6·29-s + 3·31-s − 12·35-s + 14·37-s − 2·41-s − 10·43-s + 3·47-s + 7·49-s − 22·53-s + 3·55-s − 59-s − 8·61-s + 18·65-s + 3·67-s + 10·71-s − 24·73-s − 4·77-s + 16·79-s + 4·83-s + ⋯
L(s)  = 1  − 1.34·5-s + 1.51·7-s − 0.301·11-s − 1.66·13-s + 1.94·17-s + 0.917·19-s − 0.834·23-s + 25-s + 1.11·29-s + 0.538·31-s − 2.02·35-s + 2.30·37-s − 0.312·41-s − 1.52·43-s + 0.437·47-s + 49-s − 3.02·53-s + 0.404·55-s − 0.130·59-s − 1.02·61-s + 2.23·65-s + 0.366·67-s + 1.18·71-s − 2.80·73-s − 0.455·77-s + 1.80·79-s + 0.439·83-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

\(L(1)\) \(\approx\) \(1.778070170\)
\(L(\frac12)\) \(\approx\) \(1.778070170\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
good5$C_2^2$ \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.5.d_e
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.7.ae_j
13$C_2^2$ \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.13.g_x
17$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.17.ai_by
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.19.ae_bq
23$C_2^2$ \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.23.e_ah
29$C_2^2$ \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.29.ag_h
31$C_2^2$ \( 1 - 3 T - 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.31.ad_aw
37$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.37.ao_et
41$C_2^2$ \( 1 + 2 T - 37 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.41.c_abl
43$C_2^2$ \( 1 + 10 T + 57 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.43.k_cf
47$C_2^2$ \( 1 - 3 T - 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.47.ad_abm
53$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \) 2.53.w_it
59$C_2^2$ \( 1 + T - 58 T^{2} + p T^{3} + p^{2} T^{4} \) 2.59.b_acg
61$C_2^2$ \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.61.i_d
67$C_2^2$ \( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.67.ad_acg
71$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.71.ak_gl
73$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.73.y_le
79$C_2^2$ \( 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.79.aq_gv
83$C_2^2$ \( 1 - 4 T - 67 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.83.ae_acp
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.89.au_ks
97$C_2^2$ \( 1 + 13 T + 72 T^{2} + 13 p T^{3} + p^{2} T^{4} \) 2.97.n_cu
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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

−9.366516857113474529311889257563, −8.430064337450268176001351263188, −8.295960560342039349500767244607, −7.80587013131814935041915027981, −7.75267056007100510391374930371, −7.57889826926469089522719279698, −7.10034685701693257000030637842, −6.28381258581892263470703931309, −6.25691481757592330056059904953, −5.38617739699664311405710632982, −5.10872878040690505935846385086, −4.73515862050583796451727691844, −4.58572708599440876497677357427, −3.93717170186513640406123105053, −3.46015920030183614619606681023, −2.84891690244536937420721866521, −2.68393460537481216177591709443, −1.72839494182220659120981050278, −1.25604327384858344309000904661, −0.49383437178019420489630170518, 0.49383437178019420489630170518, 1.25604327384858344309000904661, 1.72839494182220659120981050278, 2.68393460537481216177591709443, 2.84891690244536937420721866521, 3.46015920030183614619606681023, 3.93717170186513640406123105053, 4.58572708599440876497677357427, 4.73515862050583796451727691844, 5.10872878040690505935846385086, 5.38617739699664311405710632982, 6.25691481757592330056059904953, 6.28381258581892263470703931309, 7.10034685701693257000030637842, 7.57889826926469089522719279698, 7.75267056007100510391374930371, 7.80587013131814935041915027981, 8.295960560342039349500767244607, 8.430064337450268176001351263188, 9.366516857113474529311889257563

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