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Properties

Label 2-2720-85.72-c0-0-1
Degree $2$
Conductor $2720$
Sign $0.908 - 0.417i$
Analytic cond. $1.35745$
Root an. cond. $1.16509$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·5-s + 9-s + (1 − i)13-s + 17-s − 25-s + (−1 − i)29-s − 2·37-s + (1 + i)41-s + i·45-s + 49-s + (1 + i)53-s + (1 + i)61-s + (1 + i)65-s − 2i·73-s + 81-s + ⋯
L(s)  = 1  + i·5-s + 9-s + (1 − i)13-s + 17-s − 25-s + (−1 − i)29-s − 2·37-s + (1 + i)41-s + i·45-s + 49-s + (1 + i)53-s + (1 + i)61-s + (1 + i)65-s − 2i·73-s + 81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 - 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 - 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2720\)    =    \(2^{5} \cdot 5 \cdot 17\)
Sign: $0.908 - 0.417i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2720} (1857, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2720,\ (\ :0),\ 0.908 - 0.417i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.404939374\)
\(L(\frac12)\) \(\approx\) \(1.404939374\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - iT \)
17 \( 1 - T \)
good3 \( 1 - T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + iT^{2} \)
13 \( 1 + (-1 + i)T - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (1 + i)T + iT^{2} \)
31 \( 1 - iT^{2} \)
37 \( 1 + 2T + T^{2} \)
41 \( 1 + (-1 - i)T + iT^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (-1 - i)T + iT^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + (-1 - i)T + iT^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 + 2iT - T^{2} \)
79 \( 1 - iT^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - 2iT - T^{2} \)
97 \( 1 + 2T + T^{2} \)
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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

−9.155589715505141993223816173757, −8.092434028048474114462510920313, −7.56037783271574302540401520925, −6.86280728441470991658064943830, −5.98845411604239028902984261628, −5.38611241868065797509553994834, −4.05354031097896196905679301615, −3.51685098085963555151147663882, −2.50830483353623076227001140368, −1.26715928386523594445300281406, 1.19220562538788129583631887803, 1.95714710998196629949120086759, 3.65778200192572520820172544809, 4.06408123383585255785748351314, 5.14964872567129875927384556376, 5.69885941376731997094227815278, 6.84119285940158430457003546507, 7.36067437819611940183922758897, 8.398578331545730592680828757036, 8.897073763600607817927012277712

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