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Properties

Label 4-1769472-1.1-c1e2-0-37
Degree $4$
Conductor $1769472$
Sign $1$
Analytic cond. $112.823$
Root an. cond. $3.25911$
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  + 3-s + 9-s + 4·11-s + 2·17-s + 12·19-s + 2·25-s + 27-s + 4·33-s + 10·41-s − 10·49-s + 2·51-s + 12·57-s + 12·59-s − 16·67-s + 12·73-s + 2·75-s + 81-s − 4·83-s + 2·89-s + 4·99-s + 28·107-s − 6·113-s − 10·121-s + 10·123-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s + 1.20·11-s + 0.485·17-s + 2.75·19-s + 2/5·25-s + 0.192·27-s + 0.696·33-s + 1.56·41-s − 1.42·49-s + 0.280·51-s + 1.58·57-s + 1.56·59-s − 1.95·67-s + 1.40·73-s + 0.230·75-s + 1/9·81-s − 0.439·83-s + 0.211·89-s + 0.402·99-s + 2.70·107-s − 0.564·113-s − 0.909·121-s + 0.901·123-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1769472 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1769472 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1769472\)    =    \(2^{16} \cdot 3^{3}\)
Sign: $1$
Analytic conductor: \(112.823\)
Root analytic conductor: \(3.25911\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1769472,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.833989408\)
\(L(\frac12)\) \(\approx\) \(3.833989408\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.11.ae_ba
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.13.a_g
17$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) 2.17.ac_bi
19$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.19.am_cs
23$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.23.a_c
29$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.29.a_abi
31$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.31.a_c
37$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.37.a_abu
41$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.41.ak_cg
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.a_cs
47$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.47.a_bi
53$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \) 2.53.a_acw
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) 2.59.am_eo
61$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \) 2.61.a_de
67$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.67.q_ha
71$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.71.a_c
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.73.am_gk
79$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.79.a_aw
83$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.83.e_ec
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.89.ac_du
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

−7.67276898837903548687161696968, −7.53973662405198226112337542451, −7.01985631140023666271819427917, −6.67218275869700130821746322088, −6.02971056732046220646829817602, −5.76213423670232652004604237965, −5.08647324436203267742345301763, −4.84835176806430157925999829696, −4.11775929230338519280955900768, −3.71581183706563415842757588227, −3.20300572831645822036136259445, −2.89188628105598775186645079261, −2.09471433202922370037898838358, −1.29392878724921794350000779518, −0.931196891152883060137580669237, 0.931196891152883060137580669237, 1.29392878724921794350000779518, 2.09471433202922370037898838358, 2.89188628105598775186645079261, 3.20300572831645822036136259445, 3.71581183706563415842757588227, 4.11775929230338519280955900768, 4.84835176806430157925999829696, 5.08647324436203267742345301763, 5.76213423670232652004604237965, 6.02971056732046220646829817602, 6.67218275869700130821746322088, 7.01985631140023666271819427917, 7.53973662405198226112337542451, 7.67276898837903548687161696968

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