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Properties

Label 4-2352e2-1.1-c1e2-0-9
Degree $4$
Conductor $5531904$
Sign $1$
Analytic cond. $352.718$
Root an. cond. $4.33368$
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-s + 4·5-s − 4·11-s − 8·13-s + 4·15-s − 4·19-s + 5·25-s − 27-s + 4·29-s − 8·31-s − 4·33-s + 6·37-s − 8·39-s − 8·43-s + 8·47-s + 10·53-s − 16·55-s − 4·57-s − 4·59-s − 4·61-s − 32·65-s + 4·67-s − 16·71-s − 16·73-s + 5·75-s − 8·79-s − 81-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.78·5-s − 1.20·11-s − 2.21·13-s + 1.03·15-s − 0.917·19-s + 25-s − 0.192·27-s + 0.742·29-s − 1.43·31-s − 0.696·33-s + 0.986·37-s − 1.28·39-s − 1.21·43-s + 1.16·47-s + 1.37·53-s − 2.15·55-s − 0.529·57-s − 0.520·59-s − 0.512·61-s − 3.96·65-s + 0.488·67-s − 1.89·71-s − 1.87·73-s + 0.577·75-s − 0.900·79-s − 1/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5531904\)    =    \(2^{8} \cdot 3^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(352.718\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5531904,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.881416412\)
\(L(\frac12)\) \(\approx\) \(1.881416412\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
7 \( 1 \)
good5$C_2^2$ \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.5.ae_l
11$C_2^2$ \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.11.e_f
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.13.i_bq
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.17.a_ar
19$C_2^2$ \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.19.e_ad
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.23.a_ax
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.29.ae_ck
31$C_2^2$ \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.31.i_bh
37$C_2^2$ \( 1 - 6 T - T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.37.ag_ab
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$C_2^2$ \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.47.ai_r
53$C_2^2$ \( 1 - 10 T + 47 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.53.ak_bv
59$C_2^2$ \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.59.e_abr
61$C_2^2$ \( 1 + 4 T - 45 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.61.e_abt
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.67.ae_abz
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.71.q_hy
73$C_2^2$ \( 1 + 16 T + 183 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.73.q_hb
79$C_2^2$ \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.79.i_ap
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.83.y_ly
89$C_2^2$ \( 1 - 8 T - 25 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.89.ai_az
97$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.97.q_jy
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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

−9.453430356924096516954387966619, −8.774744769934464720347236597420, −8.426398773470326105187663785078, −8.204943654080108318845110275604, −7.38499561046001567850267299425, −7.30748904774936482182526723639, −7.11932411701363410314260889340, −6.38403530444109867794797676531, −5.90035329179264608648699729110, −5.63029002947119587336120004431, −5.42904588324308453994154679732, −4.74452464209366695153229308184, −4.53158274287161315924384974500, −4.02994866747904851080724044708, −3.09344413618960093487417649904, −2.73251872007533968775631724752, −2.55339438013046948776458137178, −1.81074449269045387743393060542, −1.78714223610492468647488663261, −0.40047566782599084506419003243, 0.40047566782599084506419003243, 1.78714223610492468647488663261, 1.81074449269045387743393060542, 2.55339438013046948776458137178, 2.73251872007533968775631724752, 3.09344413618960093487417649904, 4.02994866747904851080724044708, 4.53158274287161315924384974500, 4.74452464209366695153229308184, 5.42904588324308453994154679732, 5.63029002947119587336120004431, 5.90035329179264608648699729110, 6.38403530444109867794797676531, 7.11932411701363410314260889340, 7.30748904774936482182526723639, 7.38499561046001567850267299425, 8.204943654080108318845110275604, 8.426398773470326105187663785078, 8.774744769934464720347236597420, 9.453430356924096516954387966619

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