Properties

Degree $4$
Conductor $1863225$
Sign $1$
Motivic weight $1$
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 − 4-s + 8·8-s + 9-s + 8·11-s − 7·16-s − 2·18-s − 16·22-s + 16·23-s + 25-s − 4·29-s − 14·32-s − 36-s + 12·37-s − 8·43-s − 8·44-s − 32·46-s − 7·49-s − 2·50-s + 12·53-s + 8·58-s + 35·64-s − 8·67-s + 8·72-s − 24·74-s + 32·79-s + 81-s + ⋯
L(s)  = 1  − 1.41·2-s − 1/2·4-s + 2.82·8-s + 1/3·9-s + 2.41·11-s − 7/4·16-s − 0.471·18-s − 3.41·22-s + 3.33·23-s + 1/5·25-s − 0.742·29-s − 2.47·32-s − 1/6·36-s + 1.97·37-s − 1.21·43-s − 1.20·44-s − 4.71·46-s − 49-s − 0.282·50-s + 1.64·53-s + 1.05·58-s + 35/8·64-s − 0.977·67-s + 0.942·72-s − 2.78·74-s + 3.60·79-s + 1/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1863225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1863225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1863225\)    =    \(3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{1863225} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1863225,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.120692382\)
\(L(\frac12)\) \(\approx\) \(1.120692382\)
\(L(\frac{3}{2})\) 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$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_2$ \( 1 + p T^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good2$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−7.66513698740624617246413812906, −7.66093767427808108694408525270, −7.08802589844853263958022569329, −6.54785831295780264103860348787, −6.48654966834348628333399350819, −5.57900643282056552246174341208, −4.88823595164803979266926983864, −4.87469251592617747564502770211, −4.27586819251074296637919964352, −3.61843511693942583930939957195, −3.56799192480000330057521215275, −2.49981243316321035709858596980, −1.53827946418072491307494025034, −1.16776210284876177743507098510, −0.73264725064818620387577910362, 0.73264725064818620387577910362, 1.16776210284876177743507098510, 1.53827946418072491307494025034, 2.49981243316321035709858596980, 3.56799192480000330057521215275, 3.61843511693942583930939957195, 4.27586819251074296637919964352, 4.87469251592617747564502770211, 4.88823595164803979266926983864, 5.57900643282056552246174341208, 6.48654966834348628333399350819, 6.54785831295780264103860348787, 7.08802589844853263958022569329, 7.66093767427808108694408525270, 7.66513698740624617246413812906

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