Properties

Label 2-8280-1.1-c1-0-10
Degree $2$
Conductor $8280$
Sign $1$
Analytic cond. $66.1161$
Root an. cond. $8.13118$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 2·7-s − 2·13-s − 6·17-s + 23-s + 25-s + 2·35-s − 8·37-s + 12·41-s − 2·43-s − 8·47-s − 3·49-s + 10·53-s − 14·59-s + 2·61-s + 2·65-s + 2·67-s − 2·71-s − 6·73-s + 4·83-s + 6·85-s + 10·89-s + 4·91-s − 12·97-s + 14·103-s − 4·107-s + 18·109-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.755·7-s − 0.554·13-s − 1.45·17-s + 0.208·23-s + 1/5·25-s + 0.338·35-s − 1.31·37-s + 1.87·41-s − 0.304·43-s − 1.16·47-s − 3/7·49-s + 1.37·53-s − 1.82·59-s + 0.256·61-s + 0.248·65-s + 0.244·67-s − 0.237·71-s − 0.702·73-s + 0.439·83-s + 0.650·85-s + 1.05·89-s + 0.419·91-s − 1.21·97-s + 1.37·103-s − 0.386·107-s + 1.72·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8280\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $1$
Analytic conductor: \(66.1161\)
Root analytic conductor: \(8.13118\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{8280} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8280,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9120276995\)
\(L(\frac12)\) \(\approx\) \(0.9120276995\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
23 \( 1 - T \)
good7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−7.72731399349194171633351700328, −7.05811930419239877686318829119, −6.54989325780027366059452350251, −5.81809669901073407894839535109, −4.88018072312577552211371275346, −4.31633442111306918700954283314, −3.46109277812552102231847707648, −2.73712448173452933626651442096, −1.84632565661201687966394322473, −0.44925806807779103234323865527, 0.44925806807779103234323865527, 1.84632565661201687966394322473, 2.73712448173452933626651442096, 3.46109277812552102231847707648, 4.31633442111306918700954283314, 4.88018072312577552211371275346, 5.81809669901073407894839535109, 6.54989325780027366059452350251, 7.05811930419239877686318829119, 7.72731399349194171633351700328

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