Properties

Label 4-1815e2-1.1-c3e2-0-2
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 − 4·4-s − 10·5-s + 27·9-s − 24·12-s − 60·15-s − 48·16-s + 40·20-s − 150·23-s + 75·25-s + 108·27-s − 526·31-s − 108·36-s − 616·37-s − 270·45-s + 186·47-s − 288·48-s + 82·49-s + 1.05e3·53-s + 996·59-s + 240·60-s + 448·64-s + 632·67-s − 900·69-s − 576·71-s + 450·75-s + 480·80-s + ⋯
L(s)  = 1  + 1.15·3-s − 1/2·4-s − 0.894·5-s + 9-s − 0.577·12-s − 1.03·15-s − 3/4·16-s + 0.447·20-s − 1.35·23-s + 3/5·25-s + 0.769·27-s − 3.04·31-s − 1/2·36-s − 2.73·37-s − 0.894·45-s + 0.577·47-s − 0.866·48-s + 0.239·49-s + 2.72·53-s + 2.19·59-s + 0.516·60-s + 7/8·64-s + 1.15·67-s − 1.57·69-s − 0.962·71-s + 0.692·75-s + 0.670·80-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$C_2^2$ \( 1 + p^{2} T^{2} + p^{6} T^{4} \)
7$C_2^2$ \( 1 - 82 T^{2} + p^{6} T^{4} \)
13$C_2^2$ \( 1 + 926 T^{2} + p^{6} T^{4} \)
17$C_2^2$ \( 1 + 9463 T^{2} + p^{6} T^{4} \)
19$C_2^2$ \( 1 + 11690 T^{2} + p^{6} T^{4} \)
23$C_2$ \( ( 1 + 75 T + p^{3} T^{2} )^{2} \)
29$C_2^2$ \( 1 + 32350 T^{2} + p^{6} T^{4} \)
31$C_2$ \( ( 1 + 263 T + p^{3} T^{2} )^{2} \)
37$C_2$ \( ( 1 + 308 T + p^{3} T^{2} )^{2} \)
41$C_2^2$ \( 1 + 111334 T^{2} + p^{6} T^{4} \)
43$C_2^2$ \( 1 + 157562 T^{2} + p^{6} T^{4} \)
47$C_2$ \( ( 1 - 93 T + p^{3} T^{2} )^{2} \)
53$C_2$ \( ( 1 - 525 T + p^{3} T^{2} )^{2} \)
59$C_2$ \( ( 1 - 498 T + p^{3} T^{2} )^{2} \)
61$C_2^2$ \( 1 + 258887 T^{2} + p^{6} T^{4} \)
67$C_2$ \( ( 1 - 316 T + p^{3} T^{2} )^{2} \)
71$C_2$ \( ( 1 + 288 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 83854 T^{2} + p^{6} T^{4} \)
79$C_2^2$ \( 1 - 308869 T^{2} + p^{6} T^{4} \)
83$C_2^2$ \( 1 + 816874 T^{2} + p^{6} T^{4} \)
89$C_2$ \( ( 1 + 180 T + p^{3} T^{2} )^{2} \)
97$C_2$ \( ( 1 + 904 T + p^{3} T^{2} )^{2} \)
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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

−8.701934162136508345247675401591, −8.481055845634871242017171902337, −7.972585256546923368305754714680, −7.57737050200561367640296464145, −7.07385951830119899125341657325, −7.06031645382877851742609598208, −6.56058481754417684266076064664, −5.69466398987297751106240968308, −5.37155222370367159050899813474, −5.10526581403036087590839088111, −4.31552042648913352660920212143, −3.93253661583584321483268229818, −3.73139154377647754815279286811, −3.48105588785586192220011544044, −2.56593384136032162932678342029, −2.22724285663359148253805578797, −1.75485770745531347933769299760, −1.00324702179408863786012740092, 0, 0, 1.00324702179408863786012740092, 1.75485770745531347933769299760, 2.22724285663359148253805578797, 2.56593384136032162932678342029, 3.48105588785586192220011544044, 3.73139154377647754815279286811, 3.93253661583584321483268229818, 4.31552042648913352660920212143, 5.10526581403036087590839088111, 5.37155222370367159050899813474, 5.69466398987297751106240968308, 6.56058481754417684266076064664, 7.06031645382877851742609598208, 7.07385951830119899125341657325, 7.57737050200561367640296464145, 7.972585256546923368305754714680, 8.481055845634871242017171902337, 8.701934162136508345247675401591

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