Properties

Label 4-1950e2-1.1-c1e2-0-28
Degree $4$
Conductor $3802500$
Sign $1$
Analytic cond. $242.450$
Root an. cond. $3.94598$
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  − 2·2-s − 2·3-s + 3·4-s + 4·6-s − 3·7-s − 4·8-s + 3·9-s + 3·11-s − 6·12-s + 2·13-s + 6·14-s + 5·16-s − 5·17-s − 6·18-s + 3·19-s + 6·21-s − 6·22-s + 8·24-s − 4·26-s − 4·27-s − 9·28-s − 31-s − 6·32-s − 6·33-s + 10·34-s + 9·36-s − 3·37-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 3/2·4-s + 1.63·6-s − 1.13·7-s − 1.41·8-s + 9-s + 0.904·11-s − 1.73·12-s + 0.554·13-s + 1.60·14-s + 5/4·16-s − 1.21·17-s − 1.41·18-s + 0.688·19-s + 1.30·21-s − 1.27·22-s + 1.63·24-s − 0.784·26-s − 0.769·27-s − 1.70·28-s − 0.179·31-s − 1.06·32-s − 1.04·33-s + 1.71·34-s + 3/2·36-s − 0.493·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3802500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(242.450\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3802500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7175397287\)
\(L(\frac12)\) \(\approx\) \(0.7175397287\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
5 \( 1 \)
13$C_1$ \( ( 1 - T )^{2} \)
good7$D_{4}$ \( 1 + 3 T + 6 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 3 T + 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 5 T + 30 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 3 T + 30 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2^2$ \( 1 + 17 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 3 T + 66 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + T + 72 T^{2} + p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 10 T + 70 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
53$D_{4}$ \( 1 - 8 T + 81 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + T + 108 T^{2} + p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 9 T + 50 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 93 T^{2} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 3 T + 134 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
79$D_{4}$ \( 1 + 5 T + 154 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 15 T + 212 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 10 T + 162 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 18 T + 234 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
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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

−9.472787860436681976465719940768, −9.134825056179554180350409789134, −8.653466377789352576075070335817, −8.385056582679060103267575204916, −7.74528552726931371157861592291, −7.43037237241838655659166488594, −6.82669867783239936253818766638, −6.76591564220039910140805632749, −6.23684722455429737381600457336, −6.19920772982649873698092748250, −5.50645854659061271551715813891, −5.12763987103426050317334715648, −4.49745469760822920228843287744, −3.95070265072329045531588778973, −3.43051010611221387236148923197, −3.10835915413609402385084218757, −1.99845233811937189852535600610, −1.98708375898045410262641921164, −0.78442020999582491445181638244, −0.63040485423437105060734933614, 0.63040485423437105060734933614, 0.78442020999582491445181638244, 1.98708375898045410262641921164, 1.99845233811937189852535600610, 3.10835915413609402385084218757, 3.43051010611221387236148923197, 3.95070265072329045531588778973, 4.49745469760822920228843287744, 5.12763987103426050317334715648, 5.50645854659061271551715813891, 6.19920772982649873698092748250, 6.23684722455429737381600457336, 6.76591564220039910140805632749, 6.82669867783239936253818766638, 7.43037237241838655659166488594, 7.74528552726931371157861592291, 8.385056582679060103267575204916, 8.653466377789352576075070335817, 9.134825056179554180350409789134, 9.472787860436681976465719940768

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