Properties

Label 4-1815e2-1.1-c1e2-0-26
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  + 3·4-s + 2·5-s − 3·9-s + 5·16-s + 6·20-s + 10·23-s + 3·25-s + 14·31-s − 9·36-s + 10·37-s − 6·45-s − 2·47-s + 4·49-s + 6·53-s + 6·59-s + 3·64-s + 4·67-s + 4·71-s + 10·80-s + 9·81-s − 18·89-s + 30·92-s + 9·100-s + 32·103-s − 26·113-s + 20·115-s + 42·124-s + ⋯
L(s)  = 1  + 3/2·4-s + 0.894·5-s − 9-s + 5/4·16-s + 1.34·20-s + 2.08·23-s + 3/5·25-s + 2.51·31-s − 3/2·36-s + 1.64·37-s − 0.894·45-s − 0.291·47-s + 4/7·49-s + 0.824·53-s + 0.781·59-s + 3/8·64-s + 0.488·67-s + 0.474·71-s + 1.11·80-s + 81-s − 1.90·89-s + 3.12·92-s + 9/10·100-s + 3.15·103-s − 2.44·113-s + 1.86·115-s + 3.77·124-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

\(L(1)\) \(\approx\) \(4.977909085\)
\(L(\frac12)\) \(\approx\) \(4.977909085\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
5$C_1$ \( ( 1 - T )^{2} \)
11 \( 1 \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
29$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 80 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2^2$ \( 1 + 24 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 18 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−7.27235551798184107567472542307, −7.12751031327383728769576247541, −6.60677053114774298869344297105, −6.21701070809685125162297414872, −6.05862890852302505081937052618, −5.56295493704910345932800593174, −4.97612336232248743235362735232, −4.79579168740900751026542052888, −3.99761165840674192147871025147, −3.33609464081708860857704629528, −2.80674013060930219623577924836, −2.54314044356119275812107443124, −2.28175397030153669658824547077, −1.27205112463003282085915637256, −0.906207109284252358515751529287, 0.906207109284252358515751529287, 1.27205112463003282085915637256, 2.28175397030153669658824547077, 2.54314044356119275812107443124, 2.80674013060930219623577924836, 3.33609464081708860857704629528, 3.99761165840674192147871025147, 4.79579168740900751026542052888, 4.97612336232248743235362735232, 5.56295493704910345932800593174, 6.05862890852302505081937052618, 6.21701070809685125162297414872, 6.60677053114774298869344297105, 7.12751031327383728769576247541, 7.27235551798184107567472542307

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