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Properties

Label 4-351e2-1.1-c1e2-0-12
Degree $4$
Conductor $123201$
Sign $1$
Analytic cond. $7.85540$
Root an. cond. $1.67414$
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  + 2·4-s + 5·13-s + 6·17-s + 3·23-s − 25-s − 6·29-s − 11·43-s + 5·49-s + 10·52-s + 6·53-s + 13·61-s − 8·64-s + 12·68-s + 7·79-s + 6·92-s − 2·100-s − 9·101-s − 20·103-s + 6·107-s + 9·113-s − 12·116-s + 5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + ⋯
L(s)  = 1  + 4-s + 1.38·13-s + 1.45·17-s + 0.625·23-s − 1/5·25-s − 1.11·29-s − 1.67·43-s + 5/7·49-s + 1.38·52-s + 0.824·53-s + 1.66·61-s − 64-s + 1.45·68-s + 0.787·79-s + 0.625·92-s − 1/5·100-s − 0.895·101-s − 1.97·103-s + 0.580·107-s + 0.846·113-s − 1.11·116-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 123201 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123201 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(123201\)    =    \(3^{6} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(7.85540\)
Root analytic conductor: \(1.67414\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 123201,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.222520154\)
\(L(\frac12)\) \(\approx\) \(2.222520154\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad3 \( 1 \)
13$C_2$ \( 1 - 5 T + p T^{2} \)
good2$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.2.a_ac
5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.5.a_b
7$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.7.a_af
11$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.11.a_af
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.17.ag_br
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.19.a_ac
23$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) 2.23.ad_bu
29$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.29.g_bf
31$C_2^2$ \( 1 + 55 T^{2} + p^{2} T^{4} \) 2.31.a_cd
37$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.37.a_bi
41$C_2^2$ \( 1 + 37 T^{2} + p^{2} T^{4} \) 2.41.a_bl
43$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.43.l_ek
47$C_2^2$ \( 1 + 55 T^{2} + p^{2} T^{4} \) 2.47.a_cd
53$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.53.ag_db
59$C_2^2$ \( 1 - 53 T^{2} + p^{2} T^{4} \) 2.59.a_acb
61$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.61.an_fo
67$C_2^2$ \( 1 - 89 T^{2} + p^{2} T^{4} \) 2.67.a_adl
71$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \) 2.71.a_adu
73$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.73.a_bi
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.79.ah_fu
83$C_2^2$ \( 1 - 125 T^{2} + p^{2} T^{4} \) 2.83.a_aev
89$C_2^2$ \( 1 + 130 T^{2} + p^{2} T^{4} \) 2.89.a_fa
97$C_2^2$ \( 1 + 157 T^{2} + p^{2} T^{4} \) 2.97.a_gb
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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

−9.466349474000735367213859654248, −8.859667189437764579804429393865, −8.437630909820465537698299327266, −7.931834433932656344240700788548, −7.40508922178844953963104120951, −6.90659647866798872099808206901, −6.53024570511816097436782339694, −5.82682463109169139427341828230, −5.54389361343331789376628525804, −4.86868560775346140545448422675, −3.87674723290093911300355556277, −3.54431460945942731863685650369, −2.81295877514050414010278625749, −1.96538807700825581207041103901, −1.16146606538996546209383406875, 1.16146606538996546209383406875, 1.96538807700825581207041103901, 2.81295877514050414010278625749, 3.54431460945942731863685650369, 3.87674723290093911300355556277, 4.86868560775346140545448422675, 5.54389361343331789376628525804, 5.82682463109169139427341828230, 6.53024570511816097436782339694, 6.90659647866798872099808206901, 7.40508922178844953963104120951, 7.931834433932656344240700788548, 8.437630909820465537698299327266, 8.859667189437764579804429393865, 9.466349474000735367213859654248

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