Properties

Label 2-2240-40.29-c1-0-14
Degree $2$
Conductor $2240$
Sign $-0.637 - 0.770i$
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  − 1.58·3-s + (1.82 + 1.29i)5-s i·7-s − 0.475·9-s + 6.23i·11-s + 2.95·13-s + (−2.89 − 2.05i)15-s + 5.70i·17-s − 6.34i·19-s + 1.58i·21-s − 2.70i·23-s + (1.64 + 4.72i)25-s + 5.52·27-s + 3.28i·29-s − 8.81·31-s + ⋯
L(s)  = 1  − 0.917·3-s + (0.814 + 0.579i)5-s − 0.377i·7-s − 0.158·9-s + 1.87i·11-s + 0.819·13-s + (−0.747 − 0.531i)15-s + 1.38i·17-s − 1.45i·19-s + 0.346i·21-s − 0.563i·23-s + (0.328 + 0.944i)25-s + 1.06·27-s + 0.610i·29-s − 1.58·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.637 - 0.770i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $-0.637 - 0.770i$
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: \((2,\ 2240,\ (\ :1/2),\ -0.637 - 0.770i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9698724789\)
\(L(\frac12)\) \(\approx\) \(0.9698724789\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.82 - 1.29i)T \)
7 \( 1 + iT \)
good3 \( 1 + 1.58T + 3T^{2} \)
11 \( 1 - 6.23iT - 11T^{2} \)
13 \( 1 - 2.95T + 13T^{2} \)
17 \( 1 - 5.70iT - 17T^{2} \)
19 \( 1 + 6.34iT - 19T^{2} \)
23 \( 1 + 2.70iT - 23T^{2} \)
29 \( 1 - 3.28iT - 29T^{2} \)
31 \( 1 + 8.81T + 31T^{2} \)
37 \( 1 - 0.261T + 37T^{2} \)
41 \( 1 + 1.95T + 41T^{2} \)
43 \( 1 + 7.41T + 43T^{2} \)
47 \( 1 - 8.29iT - 47T^{2} \)
53 \( 1 - 11.1T + 53T^{2} \)
59 \( 1 - 0.473iT - 59T^{2} \)
61 \( 1 + 10.1iT - 61T^{2} \)
67 \( 1 + 6.57T + 67T^{2} \)
71 \( 1 + 0.374T + 71T^{2} \)
73 \( 1 + 0.725iT - 73T^{2} \)
79 \( 1 + 1.28T + 79T^{2} \)
83 \( 1 - 10.3T + 83T^{2} \)
89 \( 1 + 13.6T + 89T^{2} \)
97 \( 1 + 5.65iT - 97T^{2} \)
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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

−9.392599432596661442943475051894, −8.692481144699028976375786541041, −7.50882981216395740904226941128, −6.74616876985084276717427635080, −6.34306728744865523917670248416, −5.37379314723098625723451354776, −4.71374043853184814518072549567, −3.65397674277174571645924089537, −2.39395673706885085086980908337, −1.43828058871294151975093561589, 0.38797617754010668657402217551, 1.50554996628687847038065407550, 2.89064400522355144765838665887, 3.83466722303113372535218666630, 5.21060856513593461244318332818, 5.70414646272528056693146226602, 5.97324938944740468405813996223, 7.00237564098502811657082540681, 8.308410008801315114970343012147, 8.664193345732950130524329310848

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