Properties

Degree $4$
Conductor $22127616$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 3·9-s + 8·19-s − 10·25-s − 4·27-s − 12·29-s + 16·31-s + 4·37-s + 16·47-s − 4·53-s − 16·57-s + 8·59-s + 20·75-s + 5·81-s − 24·83-s + 24·87-s − 32·93-s + 16·103-s − 12·109-s − 8·111-s + 4·113-s + 10·121-s + 127-s + 131-s + 137-s + 139-s − 32·141-s + ⋯
L(s)  = 1  − 1.15·3-s + 9-s + 1.83·19-s − 2·25-s − 0.769·27-s − 2.22·29-s + 2.87·31-s + 0.657·37-s + 2.33·47-s − 0.549·53-s − 2.11·57-s + 1.04·59-s + 2.30·75-s + 5/9·81-s − 2.63·83-s + 2.57·87-s − 3.31·93-s + 1.57·103-s − 1.14·109-s − 0.759·111-s + 0.376·113-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.69·141-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22127616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22127616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(22127616\)    =    \(2^{10} \cdot 3^{2} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{4704} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 22127616,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.458449098\)
\(L(\frac12)\) \(\approx\) \(1.458449098\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
3$C_1$ \( ( 1 + T )^{2} \)
7 \( 1 \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 146 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + p T^{2} )^{2} \)
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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

−8.365778492225087265046386464599, −7.918612022604341217808372033098, −7.73674952388092284124541673619, −7.42593339498639227761650734454, −6.90255010493920714418944634150, −6.83858616101313393427868056700, −6.00488743153619097652152661056, −5.90727828553413060683906860704, −5.70554067670325888461056346839, −5.32485468421909927535789492452, −4.74988340245842819534936233202, −4.50514575603951211609942146847, −3.93352631022221744166972365296, −3.76824603348257562168704777764, −3.14003534714179005604200288187, −2.59824645629894592600782620378, −2.16112159249811012191308944914, −1.48164055503404388658558150737, −1.01640786358436408481736587084, −0.42626342889688379898771070604, 0.42626342889688379898771070604, 1.01640786358436408481736587084, 1.48164055503404388658558150737, 2.16112159249811012191308944914, 2.59824645629894592600782620378, 3.14003534714179005604200288187, 3.76824603348257562168704777764, 3.93352631022221744166972365296, 4.50514575603951211609942146847, 4.74988340245842819534936233202, 5.32485468421909927535789492452, 5.70554067670325888461056346839, 5.90727828553413060683906860704, 6.00488743153619097652152661056, 6.83858616101313393427868056700, 6.90255010493920714418944634150, 7.42593339498639227761650734454, 7.73674952388092284124541673619, 7.918612022604341217808372033098, 8.365778492225087265046386464599

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