Properties

Label 4-1815e2-1.1-c1e2-0-6
Degree $4$
Conductor $3294225$
Sign $1$
Analytic cond. $210.042$
Root an. cond. $3.80694$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 4-s − 2·9-s − 12-s − 3·16-s + 25-s − 5·27-s + 16·31-s + 2·36-s + 16·37-s − 3·48-s + 11·49-s + 7·64-s + 10·67-s + 75-s + 81-s + 16·93-s − 20·97-s − 100-s − 8·103-s + 5·108-s + 16·111-s − 16·124-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 0.577·3-s − 1/2·4-s − 2/3·9-s − 0.288·12-s − 3/4·16-s + 1/5·25-s − 0.962·27-s + 2.87·31-s + 1/3·36-s + 2.63·37-s − 0.433·48-s + 11/7·49-s + 7/8·64-s + 1.22·67-s + 0.115·75-s + 1/9·81-s + 1.65·93-s − 2.03·97-s − 0.0999·100-s − 0.788·103-s + 0.481·108-s + 1.51·111-s − 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3294225\)    =    \(3^{2} \cdot 5^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(210.042\)
Root analytic conductor: \(3.80694\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3294225} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3294225,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.164623951\)
\(L(\frac12)\) \(\approx\) \(2.164623951\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + p T^{2} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
11 \( 1 \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
13$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 65 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 11 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 - p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 154 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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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

−7.75108669576077247940490694154, −7.10335220905415264174250705293, −6.55990036961748052535931155016, −6.42066549995926550896802673501, −5.76583639442488939580438494044, −5.51531665774837890195558230705, −4.82846362710618164796626342915, −4.49229389936302136834913318505, −4.12460282513211220929602216437, −3.64821989159717453722069347655, −2.86922427835733851086409729022, −2.61816884699586668676435962338, −2.27069159922717332225871173756, −1.20402281371637718311529494156, −0.58559455765856009673351303498, 0.58559455765856009673351303498, 1.20402281371637718311529494156, 2.27069159922717332225871173756, 2.61816884699586668676435962338, 2.86922427835733851086409729022, 3.64821989159717453722069347655, 4.12460282513211220929602216437, 4.49229389936302136834913318505, 4.82846362710618164796626342915, 5.51531665774837890195558230705, 5.76583639442488939580438494044, 6.42066549995926550896802673501, 6.55990036961748052535931155016, 7.10335220905415264174250705293, 7.75108669576077247940490694154

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