Properties

Label 4-1850e2-1.1-c1e2-0-14
Degree $4$
Conductor $3422500$
Sign $1$
Analytic cond. $218.221$
Root an. cond. $3.84347$
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 − 3·3-s + 3·4-s − 6·6-s − 2·7-s + 4·8-s + 4·9-s − 11-s − 9·12-s + 13-s − 4·14-s + 5·16-s + 12·17-s + 8·18-s + 4·19-s + 6·21-s − 2·22-s + 3·23-s − 12·24-s + 2·26-s − 6·27-s − 6·28-s + 3·29-s + 3·31-s + 6·32-s + 3·33-s + 24·34-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.73·3-s + 3/2·4-s − 2.44·6-s − 0.755·7-s + 1.41·8-s + 4/3·9-s − 0.301·11-s − 2.59·12-s + 0.277·13-s − 1.06·14-s + 5/4·16-s + 2.91·17-s + 1.88·18-s + 0.917·19-s + 1.30·21-s − 0.426·22-s + 0.625·23-s − 2.44·24-s + 0.392·26-s − 1.15·27-s − 1.13·28-s + 0.557·29-s + 0.538·31-s + 1.06·32-s + 0.522·33-s + 4.11·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3422500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3422500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3422500\)    =    \(2^{2} \cdot 5^{4} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(218.221\)
Root analytic conductor: \(3.84347\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1850} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3422500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.940028090\)
\(L(\frac12)\) \(\approx\) \(3.940028090\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
5 \( 1 \)
37$C_1$ \( ( 1 + T )^{2} \)
good3$C_4$ \( 1 + p T + 5 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + T + 19 T^{2} + p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - T + 23 T^{2} - p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$D_{4}$ \( 1 - 3 T + 19 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
29$C_4$ \( 1 - 3 T + 31 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 3 T + 61 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 9 T + 73 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 6 T + 82 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 2 T + 82 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 3 T + 43 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 11 T + 83 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
73$D_{4}$ \( 1 - 21 T + 253 T^{2} - 21 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 7 T + 11 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 20 T + 214 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 4 T + 130 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 4 T - 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
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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

−9.718053053128716410831979470838, −9.236741405170610192710233211038, −8.565804084941438614456856665404, −8.017264901818200086858293348505, −7.50572484000297588098484172036, −7.47859109841786056540465647586, −6.67180824447499145122198324379, −6.65848133781939143594087281410, −5.99754407598893985450833549105, −5.53317025922360171796568306061, −5.47882644265647658275526629298, −5.38511586585012385601700937987, −4.62675218985074459374435324413, −4.10073678074786915873811252591, −3.71827902306834164516962095765, −3.05784995778967491571105258752, −2.95939456595408408469745601679, −2.06485302664892115251413844573, −0.973898694541845788112716551287, −0.878006943511392673067401385117, 0.878006943511392673067401385117, 0.973898694541845788112716551287, 2.06485302664892115251413844573, 2.95939456595408408469745601679, 3.05784995778967491571105258752, 3.71827902306834164516962095765, 4.10073678074786915873811252591, 4.62675218985074459374435324413, 5.38511586585012385601700937987, 5.47882644265647658275526629298, 5.53317025922360171796568306061, 5.99754407598893985450833549105, 6.65848133781939143594087281410, 6.67180824447499145122198324379, 7.47859109841786056540465647586, 7.50572484000297588098484172036, 8.017264901818200086858293348505, 8.565804084941438614456856665404, 9.236741405170610192710233211038, 9.718053053128716410831979470838

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