Properties

Label 4-462e2-1.1-c1e2-0-1
Degree $4$
Conductor $213444$
Sign $1$
Analytic cond. $13.6093$
Root an. cond. $1.92070$
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 − 3-s − 3·5-s − 6-s − 5·7-s − 8-s − 3·10-s + 11-s + 12·13-s − 5·14-s + 3·15-s − 16-s + 5·17-s − 6·19-s + 5·21-s + 22-s − 5·23-s + 24-s + 5·25-s + 12·26-s + 27-s − 12·29-s + 3·30-s − 4·31-s − 33-s + 5·34-s + 15·35-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 1.34·5-s − 0.408·6-s − 1.88·7-s − 0.353·8-s − 0.948·10-s + 0.301·11-s + 3.32·13-s − 1.33·14-s + 0.774·15-s − 1/4·16-s + 1.21·17-s − 1.37·19-s + 1.09·21-s + 0.213·22-s − 1.04·23-s + 0.204·24-s + 25-s + 2.35·26-s + 0.192·27-s − 2.22·29-s + 0.547·30-s − 0.718·31-s − 0.174·33-s + 0.857·34-s + 2.53·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(213444\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(13.6093\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{462} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 213444,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8363939240\)
\(L(\frac12)\) \(\approx\) \(0.8363939240\)
\(L(\frac{3}{2})\) 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)$
bad2$C_2$ \( 1 - T + T^{2} \)
3$C_2$ \( 1 + T + T^{2} \)
7$C_2$ \( 1 + 5 T + p T^{2} \)
11$C_2$ \( 1 - T + T^{2} \)
good5$C_2^2$ \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 5 T + 8 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 5 T + 2 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2^2$ \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 9 T + 34 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 2 T - 49 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 12 T + 71 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
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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

−11.20563809397101014650664970573, −11.08596382079880848787592403646, −10.57563800529633923998914339475, −9.872543175763159238176105725147, −9.677191152823600531507542156007, −8.866691796209111222041956602010, −8.415653659651287878995620351852, −8.401261182946546531249815981879, −7.44274598769626041223788770633, −7.07203850182775116138844693569, −6.19563026165788213932908796184, −6.17017495147869987393925431088, −5.93771831703104399924578753623, −5.13683434004154946567090740583, −4.17754879571862179316054462549, −3.82307835903289348310255895215, −3.41053038581358369306414975653, −3.36858210766550814086785619680, −1.80139740433306722210677912276, −0.53287437485740683105990951494, 0.53287437485740683105990951494, 1.80139740433306722210677912276, 3.36858210766550814086785619680, 3.41053038581358369306414975653, 3.82307835903289348310255895215, 4.17754879571862179316054462549, 5.13683434004154946567090740583, 5.93771831703104399924578753623, 6.17017495147869987393925431088, 6.19563026165788213932908796184, 7.07203850182775116138844693569, 7.44274598769626041223788770633, 8.401261182946546531249815981879, 8.415653659651287878995620351852, 8.866691796209111222041956602010, 9.677191152823600531507542156007, 9.872543175763159238176105725147, 10.57563800529633923998914339475, 11.08596382079880848787592403646, 11.20563809397101014650664970573

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