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Properties

Label 2-2240-140.139-c1-0-39
Degree $2$
Conductor $2240$
Sign $0.404 + 0.914i$
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  − 0.568i·3-s + (−2.03 − 0.918i)5-s + (−1.76 + 1.96i)7-s + 2.67·9-s + 0.417i·11-s − 2.13·13-s + (−0.521 + 1.15i)15-s + 0.314·17-s − 2.19·19-s + (1.11 + 1.00i)21-s − 0.992·23-s + (3.31 + 3.74i)25-s − 3.22i·27-s + 4.35·29-s + 4.22·31-s + ⋯
L(s)  = 1  − 0.328i·3-s + (−0.911 − 0.410i)5-s + (−0.667 + 0.744i)7-s + 0.892·9-s + 0.125i·11-s − 0.592·13-s + (−0.134 + 0.299i)15-s + 0.0761·17-s − 0.503·19-s + (0.244 + 0.219i)21-s − 0.206·23-s + (0.662 + 0.748i)25-s − 0.620i·27-s + 0.808·29-s + 0.759·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.404 + 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.404 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.404 + 0.914i$
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: \((2,\ 2240,\ (\ :1/2),\ 0.404 + 0.914i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.149282209\)
\(L(\frac12)\) \(\approx\) \(1.149282209\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.03 + 0.918i)T \)
7 \( 1 + (1.76 - 1.96i)T \)
good3 \( 1 + 0.568iT - 3T^{2} \)
11 \( 1 - 0.417iT - 11T^{2} \)
13 \( 1 + 2.13T + 13T^{2} \)
17 \( 1 - 0.314T + 17T^{2} \)
19 \( 1 + 2.19T + 19T^{2} \)
23 \( 1 + 0.992T + 23T^{2} \)
29 \( 1 - 4.35T + 29T^{2} \)
31 \( 1 - 4.22T + 31T^{2} \)
37 \( 1 + 6.83iT - 37T^{2} \)
41 \( 1 + 3.82iT - 41T^{2} \)
43 \( 1 - 6.04T + 43T^{2} \)
47 \( 1 - 4.98iT - 47T^{2} \)
53 \( 1 + 7.14iT - 53T^{2} \)
59 \( 1 + 9.50T + 59T^{2} \)
61 \( 1 - 0.783iT - 61T^{2} \)
67 \( 1 - 11.1T + 67T^{2} \)
71 \( 1 - 6.55iT - 71T^{2} \)
73 \( 1 - 5.11T + 73T^{2} \)
79 \( 1 + 11.4iT - 79T^{2} \)
83 \( 1 + 5.49iT - 83T^{2} \)
89 \( 1 + 12.5iT - 89T^{2} \)
97 \( 1 - 0.314T + 97T^{2} \)
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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

−8.844526313737431291173948706117, −8.092509789963828444745092733944, −7.35045170581632357193228336783, −6.69439891424240063401106394786, −5.80058675402922351968996726465, −4.75826849534116775296421195575, −4.09834380463032907356827789885, −3.03589098037994908290070232310, −1.97503898109868129276862491169, −0.51821976689306208582682564891, 0.938085131977022367419228696913, 2.61797354726552083025841928407, 3.55516840927841422638513647446, 4.26423609392139805975357198346, 4.90033433855851357280444026285, 6.36306951161483538465746707027, 6.82846837636495520788528479117, 7.63360411666363043961212048598, 8.230186388657156904708855456266, 9.340321196923106816587189757757

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