Properties

Degree $2$
Conductor $4410$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 5-s − 8-s + 10-s − 3·11-s − 5·13-s + 16-s + 6·17-s + 19-s − 20-s + 3·22-s − 3·23-s + 25-s + 5·26-s + 6·29-s + 4·31-s − 32-s − 6·34-s + 11·37-s − 38-s + 40-s + 3·41-s − 10·43-s − 3·44-s + 3·46-s + 3·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s − 0.904·11-s − 1.38·13-s + 1/4·16-s + 1.45·17-s + 0.229·19-s − 0.223·20-s + 0.639·22-s − 0.625·23-s + 1/5·25-s + 0.980·26-s + 1.11·29-s + 0.718·31-s − 0.176·32-s − 1.02·34-s + 1.80·37-s − 0.162·38-s + 0.158·40-s + 0.468·41-s − 1.52·43-s − 0.452·44-s + 0.442·46-s + 0.437·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4410\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{4410} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4410,\ (\ :1/2),\ -1)\)

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$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
good11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.25575451345190, −17.72459001339754, −16.99785674252820, −16.47876523618728, −15.95577879358098, −15.29079712132732, −14.66090809116334, −14.15714004238686, −13.20944639514915, −12.49418834273946, −11.95174308809134, −11.48142398041630, −10.47556201317974, −10.01164217279371, −9.606742991297362, −8.547886443199799, −7.858609172691743, −7.611071368680214, −6.739560250382706, −5.822921719036603, −5.070332121604424, −4.262547488937050, −3.031158572980807, −2.540880761115706, −1.182790072969475, 0, 1.182790072969475, 2.540880761115706, 3.031158572980807, 4.262547488937050, 5.070332121604424, 5.822921719036603, 6.739560250382706, 7.611071368680214, 7.858609172691743, 8.547886443199799, 9.606742991297362, 10.01164217279371, 10.47556201317974, 11.48142398041630, 11.95174308809134, 12.49418834273946, 13.20944639514915, 14.15714004238686, 14.66090809116334, 15.29079712132732, 15.95577879358098, 16.47876523618728, 16.99785674252820, 17.72459001339754, 18.25575451345190

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