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Properties

Label 2-2240-280.27-c0-0-3
Degree $2$
Conductor $2240$
Sign $0.584 - 0.811i$
Analytic cond. $1.11790$
Root an. cond. $1.05731$
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  + (0.541 + 0.541i)3-s + (0.382 + 0.923i)5-s + (0.707 − 0.707i)7-s − 0.414i·9-s + (−1.30 + 1.30i)13-s + (−0.292 + 0.707i)15-s + 0.765·19-s + 0.765·21-s + (1 + i)23-s + (−0.707 + 0.707i)25-s + (0.765 − 0.765i)27-s + (0.923 + 0.382i)35-s − 1.41·39-s + (0.382 − 0.158i)45-s − 1.00i·49-s + ⋯
L(s)  = 1  + (0.541 + 0.541i)3-s + (0.382 + 0.923i)5-s + (0.707 − 0.707i)7-s − 0.414i·9-s + (−1.30 + 1.30i)13-s + (−0.292 + 0.707i)15-s + 0.765·19-s + 0.765·21-s + (1 + i)23-s + (−0.707 + 0.707i)25-s + (0.765 − 0.765i)27-s + (0.923 + 0.382i)35-s − 1.41·39-s + (0.382 − 0.158i)45-s − 1.00i·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.584 - 0.811i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :0),\ 0.584 - 0.811i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.589248111\)
\(L(\frac12)\) \(\approx\) \(1.589248111\)
\(L(1)\) 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 \( 1 + (-0.382 - 0.923i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (-0.541 - 0.541i)T + iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (1.30 - 1.30i)T - iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 - 0.765T + T^{2} \)
23 \( 1 + (-1 - i)T + iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - 1.84T + T^{2} \)
61 \( 1 + 1.84T + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + 1.41iT - T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + 1.41T + T^{2} \)
83 \( 1 + (1.30 + 1.30i)T + iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - iT^{2} \)
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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

−9.450946064560105111330504711651, −8.781774277390403215678722917751, −7.52329170003483256238956038362, −7.22557741416280146183159714809, −6.37880188209509972697189948239, −5.21924677690347388656541874092, −4.42979253975528229453160353385, −3.56680332186116966749545166124, −2.72956185706165699918189233007, −1.61221537939947860884994446101, 1.17642132873890253635294420422, 2.31953099777082533626252163576, 2.90205732814530417752002091856, 4.55122073078837195146236052740, 5.18463852729974820670804898092, 5.68261392996768032401024514349, 7.02565382864962858572921739161, 7.73683270894336330599964260358, 8.346326227623662683975239270241, 8.877183369472363207714141319145

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