Properties

Label 4-1815e2-1.1-c3e2-0-3
Degree $4$
Conductor $3294225$
Sign $1$
Analytic cond. $11467.9$
Root an. cond. $10.3483$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·3-s − 16·4-s − 10·5-s + 27·9-s − 96·12-s − 60·15-s + 192·16-s + 160·20-s − 144·23-s + 75·25-s + 108·27-s − 40·31-s − 432·36-s + 428·37-s − 270·45-s − 528·47-s + 1.15e3·48-s + 34·49-s + 156·53-s + 960·59-s + 960·60-s − 2.04e3·64-s − 1.04e3·67-s − 864·69-s + 984·71-s + 450·75-s − 1.92e3·80-s + ⋯
L(s)  = 1  + 1.15·3-s − 2·4-s − 0.894·5-s + 9-s − 2.30·12-s − 1.03·15-s + 3·16-s + 1.78·20-s − 1.30·23-s + 3/5·25-s + 0.769·27-s − 0.231·31-s − 2·36-s + 1.90·37-s − 0.894·45-s − 1.63·47-s + 3.46·48-s + 0.0991·49-s + 0.404·53-s + 2.11·59-s + 2.06·60-s − 4·64-s − 1.91·67-s − 1.50·69-s + 1.64·71-s + 0.692·75-s − 2.68·80-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3294225\)    =    \(3^{2} \cdot 5^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(11467.9\)
Root analytic conductor: \(10.3483\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1815} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 3294225,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 - p T )^{2} \)
5$C_1$ \( ( 1 + p T )^{2} \)
11 \( 1 \)
good2$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
7$C_2^2$ \( 1 - 34 T^{2} + p^{6} T^{4} \)
13$C_2^2$ \( 1 + 3674 T^{2} + p^{6} T^{4} \)
17$C_2^2$ \( 1 + 6946 T^{2} + p^{6} T^{4} \)
19$C_2^2$ \( 1 + 7238 T^{2} + p^{6} T^{4} \)
23$C_2$ \( ( 1 + 72 T + p^{3} T^{2} )^{2} \)
29$C_2^2$ \( 1 - 38342 T^{2} + p^{6} T^{4} \)
31$C_2$ \( ( 1 + 20 T + p^{3} T^{2} )^{2} \)
37$C_2$ \( ( 1 - 214 T + p^{3} T^{2} )^{2} \)
41$C_2^2$ \( 1 + 137122 T^{2} + p^{6} T^{4} \)
43$C_2^2$ \( 1 + 100694 T^{2} + p^{6} T^{4} \)
47$C_2$ \( ( 1 + 264 T + p^{3} T^{2} )^{2} \)
53$C_2$ \( ( 1 - 78 T + p^{3} T^{2} )^{2} \)
59$C_2$ \( ( 1 - 480 T + p^{3} T^{2} )^{2} \)
61$C_2^2$ \( 1 + 442442 T^{2} + p^{6} T^{4} \)
67$C_2$ \( ( 1 + 524 T + p^{3} T^{2} )^{2} \)
71$C_2$ \( ( 1 - 492 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 103966 T^{2} + p^{6} T^{4} \)
79$C_2^2$ \( 1 + 898958 T^{2} + p^{6} T^{4} \)
83$C_2^2$ \( 1 + 795094 T^{2} + p^{6} T^{4} \)
89$C_2$ \( ( 1 + 1206 T + p^{3} T^{2} )^{2} \)
97$C_2$ \( ( 1 + 1186 T + p^{3} T^{2} )^{2} \)
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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

−8.505339789761480325570388560840, −8.396325060680453253113225068510, −8.058527055750189401894786916448, −7.88609479581138711086273790967, −7.21391746325451279632322316356, −7.04001468293727886004973629698, −6.12364602889260379701964746438, −5.96282739211297405241955261438, −5.13268025368723137465758497849, −5.04072163540495502989633507945, −4.22418549880912832545368315040, −4.20828066892846926766713439507, −3.80469700753090062589295519741, −3.38357280892382716259717909198, −2.80075488410104803892435845838, −2.30949000764317530133897246326, −1.33096072105625854039277987182, −1.03995354163611040087452610159, 0, 0, 1.03995354163611040087452610159, 1.33096072105625854039277987182, 2.30949000764317530133897246326, 2.80075488410104803892435845838, 3.38357280892382716259717909198, 3.80469700753090062589295519741, 4.20828066892846926766713439507, 4.22418549880912832545368315040, 5.04072163540495502989633507945, 5.13268025368723137465758497849, 5.96282739211297405241955261438, 6.12364602889260379701964746438, 7.04001468293727886004973629698, 7.21391746325451279632322316356, 7.88609479581138711086273790967, 8.058527055750189401894786916448, 8.396325060680453253113225068510, 8.505339789761480325570388560840

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