Properties

Label 4-1323e2-1.1-c1e2-0-14
Degree $4$
Conductor $1750329$
Sign $1$
Analytic cond. $111.602$
Root an. cond. $3.25026$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·4-s + 8·13-s + 2·19-s − 4·25-s + 14·31-s + 16·37-s − 2·43-s + 16·52-s − 10·61-s − 8·64-s + 4·67-s + 2·73-s + 4·76-s − 8·79-s + 2·97-s − 8·100-s − 4·103-s − 2·109-s + 2·121-s + 28·124-s + 127-s + 131-s + 137-s + 139-s + 32·148-s + 149-s + 151-s + ⋯
L(s)  = 1  + 4-s + 2.21·13-s + 0.458·19-s − 4/5·25-s + 2.51·31-s + 2.63·37-s − 0.304·43-s + 2.21·52-s − 1.28·61-s − 64-s + 0.488·67-s + 0.234·73-s + 0.458·76-s − 0.900·79-s + 0.203·97-s − 4/5·100-s − 0.394·103-s − 0.191·109-s + 2/11·121-s + 2.51·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.63·148-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1750329 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1750329 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1750329\)    =    \(3^{6} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(111.602\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1750329,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.684094338\)
\(L(\frac12)\) \(\approx\) \(3.684094338\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
7 \( 1 \)
good2$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 88 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 100 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 172 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
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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

−9.704576184006212401406534391892, −9.659799394167776585919234634684, −8.858517335179421661812818226589, −8.658548834767622359693356785920, −8.171208078533122140812875556216, −7.75619633321026076769561160862, −7.54248163045900977711521273669, −6.84764811184588095689576540153, −6.35256073278370905685106600630, −6.22286502589896433105920577290, −5.96525296975031089119311462887, −5.31807735862081517667126900023, −4.65185158550124614932619315480, −4.20073293352396634439772356827, −3.82589597855794928672562641004, −3.01162930119869255069615843900, −2.85228745878207005705032591050, −2.10649687865595426246430127606, −1.38682798543293263600778396077, −0.876824984314849652791927170507, 0.876824984314849652791927170507, 1.38682798543293263600778396077, 2.10649687865595426246430127606, 2.85228745878207005705032591050, 3.01162930119869255069615843900, 3.82589597855794928672562641004, 4.20073293352396634439772356827, 4.65185158550124614932619315480, 5.31807735862081517667126900023, 5.96525296975031089119311462887, 6.22286502589896433105920577290, 6.35256073278370905685106600630, 6.84764811184588095689576540153, 7.54248163045900977711521273669, 7.75619633321026076769561160862, 8.171208078533122140812875556216, 8.658548834767622359693356785920, 8.858517335179421661812818226589, 9.659799394167776585919234634684, 9.704576184006212401406534391892

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