Properties

Label 2-2240-140.139-c1-0-65
Degree $2$
Conductor $2240$
Sign $i$
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  − 2.64i·3-s + 2.23·5-s − 2.64i·7-s − 4.00·9-s + 5.91i·11-s + 6.70·13-s − 5.91i·15-s + 2.23·17-s − 7.00·21-s + 5.00·25-s + 2.64i·27-s + 9·29-s + 15.6·33-s − 5.91i·35-s − 17.7i·39-s + ⋯
L(s)  = 1  − 1.52i·3-s + 0.999·5-s − 0.999i·7-s − 1.33·9-s + 1.78i·11-s + 1.86·13-s − 1.52i·15-s + 0.542·17-s − 1.52·21-s + 1.00·25-s + 0.509i·27-s + 1.67·29-s + 2.72·33-s − 0.999i·35-s − 2.84i·39-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $i$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (2239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.471687406\)
\(L(\frac12)\) \(\approx\) \(2.471687406\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.23T \)
7 \( 1 + 2.64iT \)
good3 \( 1 + 2.64iT - 3T^{2} \)
11 \( 1 - 5.91iT - 11T^{2} \)
13 \( 1 - 6.70T + 13T^{2} \)
17 \( 1 - 2.23T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 9T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 7.93iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 11.8iT - 71T^{2} \)
73 \( 1 + 13.4T + 73T^{2} \)
79 \( 1 - 17.7iT - 79T^{2} \)
83 \( 1 - 15.8iT - 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + 6.70T + 97T^{2} \)
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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

−8.626306891505046435583516114331, −7.969079206824635799565574942351, −7.08003539853234198774804706796, −6.68013110300086591175341991323, −6.01998656101345551648567980017, −4.97022219418199792619987377017, −3.90622143149486664415281225481, −2.63373764384833067363313644719, −1.59798798006578019082320099172, −1.09084349034119661456841205307, 1.24398580184555648022699895802, 2.89924399571129366147212875394, 3.32410321267246777402050249021, 4.42838289832975287294250564108, 5.42659105280613253975064734232, 5.91471758231783086853878811575, 6.37699248845258382506900904825, 8.185581312362562563975621599331, 8.803905465785640655266937253567, 9.039538471472936217651485159198

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