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Properties

Label 2-280-280.69-c2-0-45
Degree $2$
Conductor $280$
Sign $0.0509 + 0.998i$
Analytic cond. $7.62944$
Root an. cond. $2.76214$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.94 − 0.448i)2-s − 3.68i·3-s + (3.59 + 1.75i)4-s + (−0.693 + 4.95i)5-s + (−1.65 + 7.18i)6-s + (4.51 − 5.34i)7-s + (−6.22 − 5.02i)8-s − 4.60·9-s + (3.57 − 9.33i)10-s + 10.2i·11-s + (6.45 − 13.2i)12-s − 14.4i·13-s + (−11.2 + 8.39i)14-s + (18.2 + 2.55i)15-s + (9.87 + 12.5i)16-s + 16.4·17-s + ⋯
L(s)  = 1  + (−0.974 − 0.224i)2-s − 1.22i·3-s + (0.899 + 0.437i)4-s + (−0.138 + 0.990i)5-s + (−0.276 + 1.19i)6-s + (0.645 − 0.763i)7-s + (−0.778 − 0.628i)8-s − 0.512·9-s + (0.357 − 0.933i)10-s + 0.928i·11-s + (0.538 − 1.10i)12-s − 1.11i·13-s + (−0.800 + 0.599i)14-s + (1.21 + 0.170i)15-s + (0.617 + 0.786i)16-s + 0.967·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0509 + 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0509 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.0509 + 0.998i$
Analytic conductor: \(7.62944\)
Root analytic conductor: \(2.76214\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1),\ 0.0509 + 0.998i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.806898 - 0.766811i\)
\(L(\frac12)\) \(\approx\) \(0.806898 - 0.766811i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.94 + 0.448i)T \)
5 \( 1 + (0.693 - 4.95i)T \)
7 \( 1 + (-4.51 + 5.34i)T \)
good3 \( 1 + 3.68iT - 9T^{2} \)
11 \( 1 - 10.2iT - 121T^{2} \)
13 \( 1 + 14.4iT - 169T^{2} \)
17 \( 1 - 16.4T + 289T^{2} \)
19 \( 1 - 31.0T + 361T^{2} \)
23 \( 1 - 20.2iT - 529T^{2} \)
29 \( 1 + 56.8iT - 841T^{2} \)
31 \( 1 + 25.7iT - 961T^{2} \)
37 \( 1 + 66.0T + 1.36e3T^{2} \)
41 \( 1 - 0.504iT - 1.68e3T^{2} \)
43 \( 1 - 26.2T + 1.84e3T^{2} \)
47 \( 1 + 20.2T + 2.20e3T^{2} \)
53 \( 1 - 53.0T + 2.80e3T^{2} \)
59 \( 1 - 59.3T + 3.48e3T^{2} \)
61 \( 1 - 69.5T + 3.72e3T^{2} \)
67 \( 1 + 61.9T + 4.48e3T^{2} \)
71 \( 1 + 25.5T + 5.04e3T^{2} \)
73 \( 1 + 34.7T + 5.32e3T^{2} \)
79 \( 1 - 64.4T + 6.24e3T^{2} \)
83 \( 1 - 9.27iT - 6.88e3T^{2} \)
89 \( 1 - 22.0iT - 7.92e3T^{2} \)
97 \( 1 - 74.3T + 9.40e3T^{2} \)
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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

−11.57110869725903012193195427522, −10.27658415480106975833585778526, −9.841391017489282173570114989692, −8.001713534251036022365865824865, −7.54154583607899527625078227205, −7.08362556483962094856817534586, −5.74940540870327567972767490731, −3.55184983428322340816294093278, −2.18283968720341776274048617657, −0.892833775161146800943570277449, 1.37210240705064904384898864580, 3.36843245510485633834042117173, 4.99674323932899260012238980223, 5.55320829330394896902042716703, 7.20847047762208475764343102032, 8.612287779189331100046578639733, 8.846949998497434847219822282884, 9.772577053898500268578031770580, 10.73358523407809147825261139591, 11.66701963538772671658492385946

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