Properties

Label 2-2240-16.5-c1-0-9
Degree $2$
Conductor $2240$
Sign $0.311 - 0.950i$
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  + (0.839 − 0.839i)3-s + (−0.707 − 0.707i)5-s i·7-s + 1.59i·9-s + (2.36 + 2.36i)11-s + (0.604 − 0.604i)13-s − 1.18·15-s − 3.62·17-s + (−5.04 + 5.04i)19-s + (−0.839 − 0.839i)21-s + 7.53i·23-s + 1.00i·25-s + (3.85 + 3.85i)27-s + (−4.76 + 4.76i)29-s − 3.76·31-s + ⋯
L(s)  = 1  + (0.484 − 0.484i)3-s + (−0.316 − 0.316i)5-s − 0.377i·7-s + 0.530i·9-s + (0.711 + 0.711i)11-s + (0.167 − 0.167i)13-s − 0.306·15-s − 0.878·17-s + (−1.15 + 1.15i)19-s + (−0.183 − 0.183i)21-s + 1.57i·23-s + 0.200i·25-s + (0.741 + 0.741i)27-s + (−0.884 + 0.884i)29-s − 0.676·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.311 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.311 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.311 - 0.950i$
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: \((2,\ 2240,\ (\ :1/2),\ 0.311 - 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.397344618\)
\(L(\frac12)\) \(\approx\) \(1.397344618\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.839 + 0.839i)T - 3iT^{2} \)
11 \( 1 + (-2.36 - 2.36i)T + 11iT^{2} \)
13 \( 1 + (-0.604 + 0.604i)T - 13iT^{2} \)
17 \( 1 + 3.62T + 17T^{2} \)
19 \( 1 + (5.04 - 5.04i)T - 19iT^{2} \)
23 \( 1 - 7.53iT - 23T^{2} \)
29 \( 1 + (4.76 - 4.76i)T - 29iT^{2} \)
31 \( 1 + 3.76T + 31T^{2} \)
37 \( 1 + (0.829 + 0.829i)T + 37iT^{2} \)
41 \( 1 + 5.50iT - 41T^{2} \)
43 \( 1 + (-7.86 - 7.86i)T + 43iT^{2} \)
47 \( 1 - 10.2T + 47T^{2} \)
53 \( 1 + (8.17 + 8.17i)T + 53iT^{2} \)
59 \( 1 + (4.06 + 4.06i)T + 59iT^{2} \)
61 \( 1 + (-6.88 + 6.88i)T - 61iT^{2} \)
67 \( 1 + (9.89 - 9.89i)T - 67iT^{2} \)
71 \( 1 + 5.93iT - 71T^{2} \)
73 \( 1 - 14.9iT - 73T^{2} \)
79 \( 1 - 5.60T + 79T^{2} \)
83 \( 1 + (7.06 - 7.06i)T - 83iT^{2} \)
89 \( 1 + 5.86iT - 89T^{2} \)
97 \( 1 - 7.10T + 97T^{2} \)
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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

−9.049052596549339889533136743316, −8.403710986831915124127016543928, −7.53434982351040956252151678104, −7.17623169678498794314792514268, −6.13994130459412782170875709393, −5.18321176390920225801484956772, −4.19295292082300594341718062274, −3.56550440949090242609328655501, −2.14319810211671483268257168138, −1.46064099030204202222506188198, 0.44286645580050718670965454735, 2.22928952595854994594761042496, 3.04243069501267577291550640554, 4.10748507225172134637882357556, 4.47406084440935583278125976548, 5.98481353309564468662126698125, 6.43822924770220578696974429430, 7.29969400261471053634338886836, 8.413391155808616945053591434989, 9.024326712773605710334657601521

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