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Properties

Label 2-280-40.29-c1-0-32
Degree $2$
Conductor $280$
Sign $0.321 + 0.946i$
Analytic cond. $2.23581$
Root an. cond. $1.49526$
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  + (0.752 − 1.19i)2-s + 2.12·3-s + (−0.867 − 1.80i)4-s + (2.02 − 0.952i)5-s + (1.59 − 2.54i)6-s + i·7-s + (−2.81 − 0.317i)8-s + 1.50·9-s + (0.381 − 3.13i)10-s + 5.19i·11-s + (−1.84 − 3.82i)12-s − 6.02·13-s + (1.19 + 0.752i)14-s + (4.29 − 2.02i)15-s + (−2.49 + 3.12i)16-s + 2.84i·17-s + ⋯
L(s)  = 1  + (0.532 − 0.846i)2-s + 1.22·3-s + (−0.433 − 0.901i)4-s + (0.904 − 0.426i)5-s + (0.651 − 1.03i)6-s + 0.377i·7-s + (−0.993 − 0.112i)8-s + 0.500·9-s + (0.120 − 0.992i)10-s + 1.56i·11-s + (−0.531 − 1.10i)12-s − 1.67·13-s + (0.320 + 0.201i)14-s + (1.10 − 0.521i)15-s + (−0.623 + 0.781i)16-s + 0.690i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.321 + 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.321 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.321 + 0.946i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.321 + 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.90803 - 1.36657i\)
\(L(\frac12)\) \(\approx\) \(1.90803 - 1.36657i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.752 + 1.19i)T \)
5 \( 1 + (-2.02 + 0.952i)T \)
7 \( 1 - iT \)
good3 \( 1 - 2.12T + 3T^{2} \)
11 \( 1 - 5.19iT - 11T^{2} \)
13 \( 1 + 6.02T + 13T^{2} \)
17 \( 1 - 2.84iT - 17T^{2} \)
19 \( 1 + 1.52iT - 19T^{2} \)
23 \( 1 + 4.80iT - 23T^{2} \)
29 \( 1 + 3.04iT - 29T^{2} \)
31 \( 1 - 5.62T + 31T^{2} \)
37 \( 1 + 4.69T + 37T^{2} \)
41 \( 1 - 5.14T + 41T^{2} \)
43 \( 1 - 5.28T + 43T^{2} \)
47 \( 1 + 6.17iT - 47T^{2} \)
53 \( 1 + 10.4T + 53T^{2} \)
59 \( 1 - 0.438iT - 59T^{2} \)
61 \( 1 + 0.0169iT - 61T^{2} \)
67 \( 1 - 3.18T + 67T^{2} \)
71 \( 1 - 5.46T + 71T^{2} \)
73 \( 1 - 6.46iT - 73T^{2} \)
79 \( 1 - 10.3T + 79T^{2} \)
83 \( 1 - 13.8T + 83T^{2} \)
89 \( 1 + 6.30T + 89T^{2} \)
97 \( 1 + 3.58iT - 97T^{2} \)
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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

−12.14121960029263665862140048981, −10.47040240486998571407485681204, −9.665438745852559253580278035762, −9.240913195687501836009223395991, −8.072950583694136514021787641125, −6.63448534355511691833028852914, −5.18957253386223770578254441470, −4.33124418167416941524976110015, −2.58601840352888663420070680856, −2.08390444220355270893972726435, 2.64056667484489876932175550754, 3.42507437581637453279124091026, 5.03250100355411786030759281831, 6.09714914691579547676349889341, 7.26694119325834469738145789411, 8.015389169723682898915623927787, 9.118683046351857571372591585261, 9.717468732198267918302262861195, 11.13068203686539444939393331706, 12.40877998646630028355145419206

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