Properties

Label 4-1815e2-1.1-c1e2-0-20
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 2·5-s + 9-s − 3·16-s + 2·20-s − 4·23-s + 3·25-s − 4·31-s − 36-s + 8·37-s − 2·45-s − 12·47-s + 2·49-s + 8·53-s − 12·59-s + 7·64-s + 8·67-s + 6·80-s + 81-s + 8·89-s + 4·92-s − 4·97-s − 3·100-s + 16·103-s − 16·113-s + 8·115-s + 4·124-s + ⋯
L(s)  = 1  − 1/2·4-s − 0.894·5-s + 1/3·9-s − 3/4·16-s + 0.447·20-s − 0.834·23-s + 3/5·25-s − 0.718·31-s − 1/6·36-s + 1.31·37-s − 0.298·45-s − 1.75·47-s + 2/7·49-s + 1.09·53-s − 1.56·59-s + 7/8·64-s + 0.977·67-s + 0.670·80-s + 1/9·81-s + 0.847·89-s + 0.417·92-s − 0.406·97-s − 0.299·100-s + 1.57·103-s − 1.50·113-s + 0.746·115-s + 0.359·124-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: \(1\)
Selberg data: \((4,\ 3294225,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
11 \( 1 \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
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$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
61$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 114 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−7.40464040898396316573413502922, −6.91538408819636718866488331109, −6.48514474486789088379505720379, −6.18854804423263531608405590789, −5.53046691433963606696513728450, −5.15406160164655127727102809388, −4.60520906272607113551819173001, −4.33618876115865514319167125330, −3.89874688556222838492530649440, −3.46758282411430900411106446660, −2.89966866495590632789433482853, −2.25388240613840196989583496896, −1.68018573414219972802205623814, −0.78217883089474894796194859274, 0, 0.78217883089474894796194859274, 1.68018573414219972802205623814, 2.25388240613840196989583496896, 2.89966866495590632789433482853, 3.46758282411430900411106446660, 3.89874688556222838492530649440, 4.33618876115865514319167125330, 4.60520906272607113551819173001, 5.15406160164655127727102809388, 5.53046691433963606696513728450, 6.18854804423263531608405590789, 6.48514474486789088379505720379, 6.91538408819636718866488331109, 7.40464040898396316573413502922

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