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Properties

Label 2-2240-40.29-c1-0-29
Degree $2$
Conductor $2240$
Sign $0.909 - 0.415i$
Analytic cond. $17.8864$
Root an. cond. $4.22924$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.09·3-s + (−2.20 + 0.370i)5-s + i·7-s − 1.80·9-s − 1.07i·11-s + 3.77·13-s + (−2.41 + 0.405i)15-s − 5.08i·17-s − 0.279i·19-s + 1.09i·21-s + 4.13i·23-s + (4.72 − 1.63i)25-s − 5.25·27-s + 4.87i·29-s + 10.1·31-s + ⋯
L(s)  = 1  + 0.631·3-s + (−0.986 + 0.165i)5-s + 0.377i·7-s − 0.601·9-s − 0.323i·11-s + 1.04·13-s + (−0.622 + 0.104i)15-s − 1.23i·17-s − 0.0641i·19-s + 0.238i·21-s + 0.861i·23-s + (0.944 − 0.327i)25-s − 1.01·27-s + 0.904i·29-s + 1.81·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.415i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.909 - 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.909 - 0.415i$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (1569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ 0.909 - 0.415i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.725037319\)
\(L(\frac12)\) \(\approx\) \(1.725037319\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (2.20 - 0.370i)T \)
7 \( 1 - iT \)
good3 \( 1 - 1.09T + 3T^{2} \)
11 \( 1 + 1.07iT - 11T^{2} \)
13 \( 1 - 3.77T + 13T^{2} \)
17 \( 1 + 5.08iT - 17T^{2} \)
19 \( 1 + 0.279iT - 19T^{2} \)
23 \( 1 - 4.13iT - 23T^{2} \)
29 \( 1 - 4.87iT - 29T^{2} \)
31 \( 1 - 10.1T + 31T^{2} \)
37 \( 1 + 2.19T + 37T^{2} \)
41 \( 1 - 11.0T + 41T^{2} \)
43 \( 1 - 1.86T + 43T^{2} \)
47 \( 1 - 4.78iT - 47T^{2} \)
53 \( 1 + 5.30T + 53T^{2} \)
59 \( 1 + 5.22iT - 59T^{2} \)
61 \( 1 - 14.5iT - 61T^{2} \)
67 \( 1 - 9.74T + 67T^{2} \)
71 \( 1 - 4.18T + 71T^{2} \)
73 \( 1 - 12.4iT - 73T^{2} \)
79 \( 1 - 12.4T + 79T^{2} \)
83 \( 1 - 2.13T + 83T^{2} \)
89 \( 1 - 8.20T + 89T^{2} \)
97 \( 1 - 9.92iT - 97T^{2} \)
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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

−8.972455093981332806390600834426, −8.288610116535643612764438798135, −7.77195662531033612268738559411, −6.87057680745703219047242115504, −5.97726004680996283395135936375, −5.07464190943264554563156861247, −4.03328930464300501995282366716, −3.19988870817953300948834459433, −2.60252411693239390291531273359, −0.926253094389773005173643582666, 0.75089290899462345970497408469, 2.21140775330831843652679613420, 3.33234982003384332955897811596, 3.97192583220424390333704690258, 4.73302807976225615766656067489, 6.01890050693203101730860970068, 6.61822242021735097043805411960, 7.86484550225759129221132428265, 8.101946963655664587705005351030, 8.761759714236477004083016863524

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