Properties

Label 4-6930e2-1.1-c1e2-0-7
Degree $4$
Conductor $48024900$
Sign $1$
Analytic cond. $3062.10$
Root an. cond. $7.43883$
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  − 2·2-s + 3·4-s + 2·5-s + 2·7-s − 4·8-s − 4·10-s − 2·11-s + 4·13-s − 4·14-s + 5·16-s + 10·19-s + 6·20-s + 4·22-s + 6·23-s + 3·25-s − 8·26-s + 6·28-s + 6·29-s + 4·31-s − 6·32-s + 4·35-s − 2·37-s − 20·38-s − 8·40-s − 6·41-s + 4·43-s − 6·44-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s + 0.894·5-s + 0.755·7-s − 1.41·8-s − 1.26·10-s − 0.603·11-s + 1.10·13-s − 1.06·14-s + 5/4·16-s + 2.29·19-s + 1.34·20-s + 0.852·22-s + 1.25·23-s + 3/5·25-s − 1.56·26-s + 1.13·28-s + 1.11·29-s + 0.718·31-s − 1.06·32-s + 0.676·35-s − 0.328·37-s − 3.24·38-s − 1.26·40-s − 0.937·41-s + 0.609·43-s − 0.904·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48024900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48024900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(48024900\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(3062.10\)
Root analytic conductor: \(7.43883\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{6930} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 48024900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.307190751\)
\(L(\frac12)\) \(\approx\) \(3.307190751\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
3 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
11$C_1$ \( ( 1 + T )^{2} \)
good13$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 10 T + 60 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 6 T + 64 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$D_{4}$ \( 1 + 2 T + 72 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 6 T + 64 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 18 T + 184 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 14 T + 180 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 12 T + 94 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 166 T^{2} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 2 T - 48 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
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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

−8.251938760688604696883783565364, −7.80748736770738367921748501123, −7.46718747737599704874413811137, −7.21839684304453786256179730672, −6.72832019707687907676775785760, −6.71119733089994755466345633731, −5.86688225846272254187912827572, −5.84126860775113961707004879430, −5.36326074032533073149393101568, −5.21484201824838011314627972235, −4.55389300723680193371454084661, −4.30578912311957278223543666010, −3.49208357857907319786904778050, −3.16990213817617684112609492888, −2.78966160944805548774693460026, −2.51212391152536451503767539070, −1.69057229522510183493212845516, −1.56155159404533382987889960735, −0.841297465651754172922726154384, −0.78387952088593746868922185064, 0.78387952088593746868922185064, 0.841297465651754172922726154384, 1.56155159404533382987889960735, 1.69057229522510183493212845516, 2.51212391152536451503767539070, 2.78966160944805548774693460026, 3.16990213817617684112609492888, 3.49208357857907319786904778050, 4.30578912311957278223543666010, 4.55389300723680193371454084661, 5.21484201824838011314627972235, 5.36326074032533073149393101568, 5.84126860775113961707004879430, 5.86688225846272254187912827572, 6.71119733089994755466345633731, 6.72832019707687907676775785760, 7.21839684304453786256179730672, 7.46718747737599704874413811137, 7.80748736770738367921748501123, 8.251938760688604696883783565364

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