Properties

Label 2-8820-1.1-c1-0-49
Degree $2$
Conductor $8820$
Sign $-1$
Analytic cond. $70.4280$
Root an. cond. $8.39214$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 3.82·11-s − 3.58·13-s + 6.41·17-s − 3.65·19-s − 0.585·23-s + 25-s − 6.65·29-s − 4.58·31-s − 3.41·37-s + 0.585·41-s + 11.6·43-s − 8.89·47-s + 3.75·53-s − 3.82·55-s + 3.41·59-s − 5.17·61-s + 3.58·65-s − 11.0·67-s − 6.48·71-s − 5.17·73-s + 13.1·79-s − 8·83-s − 6.41·85-s + 16.9·89-s + 3.65·95-s − 15.7·97-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.15·11-s − 0.994·13-s + 1.55·17-s − 0.838·19-s − 0.122·23-s + 0.200·25-s − 1.23·29-s − 0.823·31-s − 0.561·37-s + 0.0914·41-s + 1.77·43-s − 1.29·47-s + 0.516·53-s − 0.516·55-s + 0.444·59-s − 0.662·61-s + 0.444·65-s − 1.35·67-s − 0.769·71-s − 0.605·73-s + 1.47·79-s − 0.878·83-s − 0.695·85-s + 1.79·89-s + 0.375·95-s − 1.59·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8820\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(70.4280\)
Root analytic conductor: \(8.39214\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{8820} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8820,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
good11 \( 1 - 3.82T + 11T^{2} \)
13 \( 1 + 3.58T + 13T^{2} \)
17 \( 1 - 6.41T + 17T^{2} \)
19 \( 1 + 3.65T + 19T^{2} \)
23 \( 1 + 0.585T + 23T^{2} \)
29 \( 1 + 6.65T + 29T^{2} \)
31 \( 1 + 4.58T + 31T^{2} \)
37 \( 1 + 3.41T + 37T^{2} \)
41 \( 1 - 0.585T + 41T^{2} \)
43 \( 1 - 11.6T + 43T^{2} \)
47 \( 1 + 8.89T + 47T^{2} \)
53 \( 1 - 3.75T + 53T^{2} \)
59 \( 1 - 3.41T + 59T^{2} \)
61 \( 1 + 5.17T + 61T^{2} \)
67 \( 1 + 11.0T + 67T^{2} \)
71 \( 1 + 6.48T + 71T^{2} \)
73 \( 1 + 5.17T + 73T^{2} \)
79 \( 1 - 13.1T + 79T^{2} \)
83 \( 1 + 8T + 83T^{2} \)
89 \( 1 - 16.9T + 89T^{2} \)
97 \( 1 + 15.7T + 97T^{2} \)
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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

−7.51506839083669035790704428209, −6.82565001819282357447601997375, −6.00786672581556770267742002191, −5.38931518456093638143642162686, −4.51371035017607411203142572783, −3.84517069573274239588195828617, −3.21147413039260216725818467844, −2.14755042855335536465212613578, −1.24424481041646592185485850622, 0, 1.24424481041646592185485850622, 2.14755042855335536465212613578, 3.21147413039260216725818467844, 3.84517069573274239588195828617, 4.51371035017607411203142572783, 5.38931518456093638143642162686, 6.00786672581556770267742002191, 6.82565001819282357447601997375, 7.51506839083669035790704428209

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