Properties

Label 2-2240-40.29-c1-0-54
Degree $2$
Conductor $2240$
Sign $-0.983 + 0.178i$
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  − 2.05·3-s + (1.27 − 1.83i)5-s i·7-s + 1.24·9-s + 5.33i·11-s − 0.785·13-s + (−2.62 + 3.78i)15-s − 1.61i·17-s + 4.54i·19-s + 2.05i·21-s − 7.09i·23-s + (−1.75 − 4.68i)25-s + 3.62·27-s − 7.74i·29-s − 4.11·31-s + ⋯
L(s)  = 1  − 1.18·3-s + (0.569 − 0.822i)5-s − 0.377i·7-s + 0.413·9-s + 1.60i·11-s − 0.217·13-s + (−0.676 + 0.977i)15-s − 0.392i·17-s + 1.04i·19-s + 0.449i·21-s − 1.47i·23-s + (−0.351 − 0.936i)25-s + 0.697·27-s − 1.43i·29-s − 0.739·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.178i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 + 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $-0.983 + 0.178i$
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: \((2,\ 2240,\ (\ :1/2),\ -0.983 + 0.178i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3283738596\)
\(L(\frac12)\) \(\approx\) \(0.3283738596\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.27 + 1.83i)T \)
7 \( 1 + iT \)
good3 \( 1 + 2.05T + 3T^{2} \)
11 \( 1 - 5.33iT - 11T^{2} \)
13 \( 1 + 0.785T + 13T^{2} \)
17 \( 1 + 1.61iT - 17T^{2} \)
19 \( 1 - 4.54iT - 19T^{2} \)
23 \( 1 + 7.09iT - 23T^{2} \)
29 \( 1 + 7.74iT - 29T^{2} \)
31 \( 1 + 4.11T + 31T^{2} \)
37 \( 1 - 0.480T + 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 - 2.00T + 43T^{2} \)
47 \( 1 + 2.24iT - 47T^{2} \)
53 \( 1 + 5.57T + 53T^{2} \)
59 \( 1 - 5.02iT - 59T^{2} \)
61 \( 1 - 4.55iT - 61T^{2} \)
67 \( 1 + 12.5T + 67T^{2} \)
71 \( 1 + 13.4T + 71T^{2} \)
73 \( 1 - 2.00iT - 73T^{2} \)
79 \( 1 - 6.61T + 79T^{2} \)
83 \( 1 + 8.91T + 83T^{2} \)
89 \( 1 - 0.428T + 89T^{2} \)
97 \( 1 + 13.0iT - 97T^{2} \)
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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

−8.749622533660191848155458645218, −7.79225601164803275911050880518, −7.00784970220635947314186619174, −6.14902678827421664995167376851, −5.56359432985631750341897467990, −4.60536420072959467210410922212, −4.29636633456179820817959196418, −2.49166400752802777459498586444, −1.43558303572464063652786166770, −0.13842343463171370599553334218, 1.38061566956664894526737355602, 2.80586138128467381456020987468, 3.51420557382950407331000258896, 4.97801319344418580124860381732, 5.63265034414492312139408090991, 6.10927554230197636711826407087, 6.83990589794744086134785499979, 7.68564859388517210526887296217, 8.820412911762690319742551536795, 9.355430296585916304971577228997

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