Properties

Label 4-1305e2-1.1-c1e2-0-13
Degree $4$
Conductor $1703025$
Sign $1$
Analytic cond. $108.586$
Root an. cond. $3.22807$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2·4-s + 2·5-s − 6·7-s − 3·8-s + 2·10-s + 2·11-s − 8·13-s − 6·14-s + 16-s + 2·17-s + 2·19-s − 4·20-s + 2·22-s + 3·25-s − 8·26-s + 12·28-s − 2·29-s − 16·31-s + 2·32-s + 2·34-s − 12·35-s − 16·37-s + 2·38-s − 6·40-s + 2·43-s − 4·44-s + ⋯
L(s)  = 1  + 0.707·2-s − 4-s + 0.894·5-s − 2.26·7-s − 1.06·8-s + 0.632·10-s + 0.603·11-s − 2.21·13-s − 1.60·14-s + 1/4·16-s + 0.485·17-s + 0.458·19-s − 0.894·20-s + 0.426·22-s + 3/5·25-s − 1.56·26-s + 2.26·28-s − 0.371·29-s − 2.87·31-s + 0.353·32-s + 0.342·34-s − 2.02·35-s − 2.63·37-s + 0.324·38-s − 0.948·40-s + 0.304·43-s − 0.603·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1703025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1703025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1703025\)    =    \(3^{4} \cdot 5^{2} \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(108.586\)
Root analytic conductor: \(3.22807\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1703025,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
29$C_1$ \( ( 1 + T )^{2} \)
good2$D_{4}$ \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
11$D_{4}$ \( 1 - 2 T + 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 8 T + 37 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 2 T + 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$C_4$ \( 1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 2 T + 82 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 49 T^{2} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 18 T + 182 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 6 T + 122 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 2 T + 78 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 14 T + 163 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 10 T + 162 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
79$D_{4}$ \( 1 - 10 T + 58 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 6 T + 130 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 53 T^{2} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 10 T + 214 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
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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

−9.354696403449622946048466730611, −9.328163951011667657685019483990, −8.871522193565303301409709100664, −8.517157331709636875718521010631, −7.43220042678295583946744860646, −7.33805759860330208546512951483, −7.00431570598775719021715185636, −6.52717907393875981455482053830, −5.90487852030472070288178231106, −5.61319524016706448913866919633, −5.20504930187934825297667122058, −4.99424507940785809797463542745, −4.02203690811402652566381569593, −3.96029884260651070620463162534, −3.18858244896882346992177232948, −3.04291143435154111696576923677, −2.26582325386719369073341016964, −1.54053126746915855995385000648, 0, 0, 1.54053126746915855995385000648, 2.26582325386719369073341016964, 3.04291143435154111696576923677, 3.18858244896882346992177232948, 3.96029884260651070620463162534, 4.02203690811402652566381569593, 4.99424507940785809797463542745, 5.20504930187934825297667122058, 5.61319524016706448913866919633, 5.90487852030472070288178231106, 6.52717907393875981455482053830, 7.00431570598775719021715185636, 7.33805759860330208546512951483, 7.43220042678295583946744860646, 8.517157331709636875718521010631, 8.871522193565303301409709100664, 9.328163951011667657685019483990, 9.354696403449622946048466730611

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