Properties

Label 4-1152e2-1.1-c1e2-0-4
Degree $4$
Conductor $1327104$
Sign $1$
Analytic cond. $84.6173$
Root an. cond. $3.03294$
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  + 4·11-s − 10·17-s − 8·19-s − 2·25-s − 2·41-s − 4·43-s + 10·49-s + 4·59-s + 8·67-s + 20·73-s − 12·83-s − 18·89-s + 4·107-s + 22·113-s + 6·121-s + ⋯
L(s)  = 1  + 1.20·11-s − 2.42·17-s − 1.83·19-s − 2/5·25-s − 0.312·41-s − 0.609·43-s + 10/7·49-s + 0.520·59-s + 0.977·67-s + 2.34·73-s − 1.31·83-s − 1.90·89-s + 0.386·107-s + 2.06·113-s + 6/11·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1327104\)    =    \(2^{14} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(84.6173\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1327104,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.387040605\)
\(L(\frac12)\) \(\approx\) \(1.387040605\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
11$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.11.ae_k
13$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.13.a_s
17$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.k_cg
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.23.a_c
29$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \) 2.29.a_abm
31$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.31.a_o
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.37.a_ak
41$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.41.c_de
43$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.e_cc
47$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.47.a_c
53$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \) 2.53.a_ack
59$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.ae_di
61$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.61.a_bm
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.ai_di
71$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \) 2.71.a_adq
73$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.73.au_iw
79$C_2^2$ \( 1 + 102 T^{2} + p^{2} T^{4} \) 2.79.a_dy
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.83.m_hu
89$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.89.s_gw
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.a_ac
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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

−8.259943420020940766758184117309, −7.36095589392801303860216350586, −6.96259674218439822308575601750, −6.71897097710564911369081346153, −6.29730446641304632160757720926, −5.96788616336236092215934604907, −5.29226451398894724226372634449, −4.71950776476740124904616248098, −4.21405603014947122489160549973, −4.06801721625263657952218310226, −3.47905772404547454986265692990, −2.56170967319968585838985548155, −2.15927451199855623534985218978, −1.65525180571225702415782163509, −0.50040257359073248466280640099, 0.50040257359073248466280640099, 1.65525180571225702415782163509, 2.15927451199855623534985218978, 2.56170967319968585838985548155, 3.47905772404547454986265692990, 4.06801721625263657952218310226, 4.21405603014947122489160549973, 4.71950776476740124904616248098, 5.29226451398894724226372634449, 5.96788616336236092215934604907, 6.29730446641304632160757720926, 6.71897097710564911369081346153, 6.96259674218439822308575601750, 7.36095589392801303860216350586, 8.259943420020940766758184117309

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