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Properties

Label 4-1769472-1.1-c1e2-0-10
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  + 3·3-s + 6·9-s − 3·11-s − 9·17-s − 12·19-s + 3·25-s + 9·27-s − 9·33-s + 9·41-s + 6·43-s − 4·49-s − 27·51-s − 36·57-s + 9·59-s + 12·67-s + 12·73-s + 9·75-s + 9·81-s + 18·83-s + 9·89-s + 18·97-s − 18·99-s − 9·113-s − 13·121-s + 27·123-s + 127-s + 18·129-s + ⋯
L(s)  = 1  + 1.73·3-s + 2·9-s − 0.904·11-s − 2.18·17-s − 2.75·19-s + 3/5·25-s + 1.73·27-s − 1.56·33-s + 1.40·41-s + 0.914·43-s − 4/7·49-s − 3.78·51-s − 4.76·57-s + 1.17·59-s + 1.46·67-s + 1.40·73-s + 1.03·75-s + 81-s + 1.97·83-s + 0.953·89-s + 1.82·97-s − 1.80·99-s − 0.846·113-s − 1.18·121-s + 2.43·123-s + 0.0887·127-s + 1.58·129-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

\(L(1)\) \(\approx\) \(3.033901687\)
\(L(\frac12)\) \(\approx\) \(3.033901687\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - p T + p T^{2} \)
good5$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \) 2.5.a_ad
7$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.7.a_e
11$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.11.d_w
13$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.13.a_ai
17$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.17.j_bw
19$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.19.m_cw
23$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \) 2.23.a_ar
29$C_2^2$ \( 1 - 33 T^{2} + p^{2} T^{4} \) 2.29.a_abh
31$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \) 2.31.a_bc
37$C_2^2$ \( 1 - 52 T^{2} + p^{2} T^{4} \) 2.37.a_aca
41$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.41.aj_dm
43$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.43.ag_di
47$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.47.a_af
53$C_2^2$ \( 1 + 63 T^{2} + p^{2} T^{4} \) 2.53.a_cl
59$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) 2.59.aj_fg
61$C_2^2$ \( 1 + 68 T^{2} + p^{2} T^{4} \) 2.61.a_cq
67$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) 2.67.am_gf
71$C_2^2$ \( 1 - 65 T^{2} + p^{2} T^{4} \) 2.71.a_acn
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.73.am_ha
79$C_2^2$ \( 1 - 122 T^{2} + p^{2} T^{4} \) 2.79.a_aes
83$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.83.as_je
89$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.89.aj_he
97$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + p T^{2} ) \) 2.97.as_hm
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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

−7.890327764932980221856451659376, −7.65320354704771349155237539162, −6.83616114754457655253758739944, −6.73232626850287649483336194106, −6.34557128417490157174379700969, −5.71900219272245459035251262557, −4.91646408905165589452205009498, −4.64363732314390714339298961135, −4.07483014874599579002453126283, −3.90948811160586190962324856848, −3.14116733981956081233584954458, −2.44975145693351339309631160997, −2.23873540662771517942220044410, −1.97667173550287969679366501134, −0.60671002202829121535011501691, 0.60671002202829121535011501691, 1.97667173550287969679366501134, 2.23873540662771517942220044410, 2.44975145693351339309631160997, 3.14116733981956081233584954458, 3.90948811160586190962324856848, 4.07483014874599579002453126283, 4.64363732314390714339298961135, 4.91646408905165589452205009498, 5.71900219272245459035251262557, 6.34557128417490157174379700969, 6.73232626850287649483336194106, 6.83616114754457655253758739944, 7.65320354704771349155237539162, 7.890327764932980221856451659376

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