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Properties

Label 2-2240-280.27-c0-0-1
Degree $2$
Conductor $2240$
Sign $-0.811 - 0.584i$
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  + (1.30 + 1.30i)3-s + (−0.923 + 0.382i)5-s + (−0.707 + 0.707i)7-s + 2.41i·9-s + (0.541 − 0.541i)13-s + (−1.70 − 0.707i)15-s − 1.84·19-s − 1.84·21-s + (1 + i)23-s + (0.707 − 0.707i)25-s + (−1.84 + 1.84i)27-s + (0.382 − 0.923i)35-s + 1.41·39-s + (−0.923 − 2.23i)45-s − 1.00i·49-s + ⋯
L(s)  = 1  + (1.30 + 1.30i)3-s + (−0.923 + 0.382i)5-s + (−0.707 + 0.707i)7-s + 2.41i·9-s + (0.541 − 0.541i)13-s + (−1.70 − 0.707i)15-s − 1.84·19-s − 1.84·21-s + (1 + i)23-s + (0.707 − 0.707i)25-s + (−1.84 + 1.84i)27-s + (0.382 − 0.923i)35-s + 1.41·39-s + (−0.923 − 2.23i)45-s − 1.00i·49-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $-0.811 - 0.584i$
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: \((2,\ 2240,\ (\ :0),\ -0.811 - 0.584i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.299913762\)
\(L(\frac12)\) \(\approx\) \(1.299913762\)
\(L(1)\) 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 \( 1 + (0.923 - 0.382i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + (-1.30 - 1.30i)T + iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (-0.541 + 0.541i)T - iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + 1.84T + T^{2} \)
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 \( 1 - 0.765T + T^{2} \)
61 \( 1 + 0.765T + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - 1.41iT - T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 - 1.41T + T^{2} \)
83 \( 1 + (-0.541 - 0.541i)T + iT^{2} \)
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

−9.402615102541278455403187027448, −8.693951248127843519053595791479, −8.342716644492246904606641387978, −7.46834626830248271378361342837, −6.47595957516832662438610011733, −5.38200168368310766564988289239, −4.42855580266388480889443675880, −3.67306133540512001098154439898, −3.11539628743766851994765122986, −2.28824867090384378534073371890, 0.75865084056848908075667196757, 1.99301383785050433208954701319, 3.06406020904333071501182490465, 3.81102928708183417026827280178, 4.55384393599562230350568053030, 6.33861720427006826627566619260, 6.69780430578246008369096883173, 7.46101113317987346731773312760, 8.129331318633252075850318942233, 8.812139026822448705269698097170

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