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Properties

Label 2-2240-16.5-c1-0-7
Degree $2$
Conductor $2240$
Sign $-0.943 - 0.330i$
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  + (−2.22 + 2.22i)3-s + (0.707 + 0.707i)5-s − i·7-s − 6.85i·9-s + (1.33 + 1.33i)11-s + (0.657 − 0.657i)13-s − 3.13·15-s − 1.24·17-s + (1.56 − 1.56i)19-s + (2.22 + 2.22i)21-s + 3.20i·23-s + 1.00i·25-s + (8.56 + 8.56i)27-s + (3.83 − 3.83i)29-s − 11.1·31-s + ⋯
L(s)  = 1  + (−1.28 + 1.28i)3-s + (0.316 + 0.316i)5-s − 0.377i·7-s − 2.28i·9-s + (0.401 + 0.401i)11-s + (0.182 − 0.182i)13-s − 0.810·15-s − 0.302·17-s + (0.359 − 0.359i)19-s + (0.484 + 0.484i)21-s + 0.667i·23-s + 0.200i·25-s + (1.64 + 1.64i)27-s + (0.711 − 0.711i)29-s − 1.99·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 - 0.330i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.943 - 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $-0.943 - 0.330i$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (561, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ -0.943 - 0.330i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7850909138\)
\(L(\frac12)\) \(\approx\) \(0.7850909138\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 - 0.707i)T \)
7 \( 1 + iT \)
good3 \( 1 + (2.22 - 2.22i)T - 3iT^{2} \)
11 \( 1 + (-1.33 - 1.33i)T + 11iT^{2} \)
13 \( 1 + (-0.657 + 0.657i)T - 13iT^{2} \)
17 \( 1 + 1.24T + 17T^{2} \)
19 \( 1 + (-1.56 + 1.56i)T - 19iT^{2} \)
23 \( 1 - 3.20iT - 23T^{2} \)
29 \( 1 + (-3.83 + 3.83i)T - 29iT^{2} \)
31 \( 1 + 11.1T + 31T^{2} \)
37 \( 1 + (-4.06 - 4.06i)T + 37iT^{2} \)
41 \( 1 - 12.5iT - 41T^{2} \)
43 \( 1 + (-2.18 - 2.18i)T + 43iT^{2} \)
47 \( 1 - 1.62T + 47T^{2} \)
53 \( 1 + (-7.72 - 7.72i)T + 53iT^{2} \)
59 \( 1 + (-0.930 - 0.930i)T + 59iT^{2} \)
61 \( 1 + (6.14 - 6.14i)T - 61iT^{2} \)
67 \( 1 + (8.05 - 8.05i)T - 67iT^{2} \)
71 \( 1 + 12.1iT - 71T^{2} \)
73 \( 1 - 11.1iT - 73T^{2} \)
79 \( 1 - 9.90T + 79T^{2} \)
83 \( 1 + (4.56 - 4.56i)T - 83iT^{2} \)
89 \( 1 + 14.3iT - 89T^{2} \)
97 \( 1 + 2.65T + 97T^{2} \)
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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

−9.526026820912212531247217939411, −9.036059446009412863275402812460, −7.69908117056221309207964395026, −6.81253915935479179459673539265, −6.08494942470182564055791748525, −5.44187421496431859459601376334, −4.55052775474036136465232178238, −3.97678295118135821919779191968, −2.92856201606774272159960158280, −1.19653074109414114103631250694, 0.36555133173052336912631093114, 1.50027041341111122709469348993, 2.33439977204805152696982767057, 3.86419669208938276891727130608, 5.12877709069199701158266225806, 5.60112309064957131054195361257, 6.33836305708245026634811170388, 6.97968799067561426007652971968, 7.71469437244931012914175874405, 8.662907762694721460171046954291

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