Properties

Label 4-1305e2-1.1-c1e2-0-6
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 $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2·4-s − 2·5-s + 2·7-s + 5·8-s − 2·10-s − 10·11-s + 2·14-s + 5·16-s + 6·17-s − 2·19-s − 4·20-s − 10·22-s + 8·23-s + 3·25-s + 4·28-s − 2·29-s + 8·31-s + 10·32-s + 6·34-s − 4·35-s − 8·37-s − 2·38-s − 10·40-s − 10·43-s − 20·44-s + 8·46-s + ⋯
L(s)  = 1  + 0.707·2-s + 4-s − 0.894·5-s + 0.755·7-s + 1.76·8-s − 0.632·10-s − 3.01·11-s + 0.534·14-s + 5/4·16-s + 1.45·17-s − 0.458·19-s − 0.894·20-s − 2.13·22-s + 1.66·23-s + 3/5·25-s + 0.755·28-s − 0.371·29-s + 1.43·31-s + 1.76·32-s + 1.02·34-s − 0.676·35-s − 1.31·37-s − 0.324·38-s − 1.58·40-s − 1.52·43-s − 3.01·44-s + 1.17·46-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: \(0\)
Selberg data: \((4,\ 1703025,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.273166751\)
\(L(\frac12)\) \(\approx\) \(3.273166751\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
19$D_{4}$ \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 10 T + 90 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 12 T + 109 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 10 T + 110 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 6 T + 106 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 2 T - 66 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 10 T + 75 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 10 T + 146 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
79$D_{4}$ \( 1 - 6 T + 146 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 14 T + 194 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 12 T + 193 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 14 T + 222 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
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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

−10.15711805068897051049620413457, −9.859305826106954012877599441931, −8.767952699352663135566731424857, −8.357115316915182064836072936196, −7.903555847505768846152238391293, −7.87076498499061829011102609335, −7.66886914070631679152925175304, −6.85875844297782352079929851335, −6.79452836761544173571285866947, −6.11364945338429630048357510247, −5.20906301146983166694475064986, −5.19039359417866767465910634504, −4.84005163502576090222198072988, −4.64022693040538000446964690507, −3.59341397086996229110883014634, −3.25117926876776484171041075535, −2.89471569634958048054299088791, −2.14483649046238086318972586305, −1.66875754246049497979161296314, −0.66149352268979187435367943342, 0.66149352268979187435367943342, 1.66875754246049497979161296314, 2.14483649046238086318972586305, 2.89471569634958048054299088791, 3.25117926876776484171041075535, 3.59341397086996229110883014634, 4.64022693040538000446964690507, 4.84005163502576090222198072988, 5.19039359417866767465910634504, 5.20906301146983166694475064986, 6.11364945338429630048357510247, 6.79452836761544173571285866947, 6.85875844297782352079929851335, 7.66886914070631679152925175304, 7.87076498499061829011102609335, 7.903555847505768846152238391293, 8.357115316915182064836072936196, 8.767952699352663135566731424857, 9.859305826106954012877599441931, 10.15711805068897051049620413457

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