Properties

Label 2-280-280.19-c1-0-10
Degree $2$
Conductor $280$
Sign $0.271 - 0.962i$
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  + (−1.41 + 0.0828i)2-s + (0.653 + 1.13i)3-s + (1.98 − 0.233i)4-s + (1.67 + 1.48i)5-s + (−1.01 − 1.54i)6-s + (1.03 + 2.43i)7-s + (−2.78 + 0.494i)8-s + (0.646 − 1.11i)9-s + (−2.48 − 1.95i)10-s + (0.168 + 0.292i)11-s + (1.56 + 2.09i)12-s + 1.29i·13-s + (−1.66 − 3.35i)14-s + (−0.588 + 2.86i)15-s + (3.89 − 0.929i)16-s + (−3.41 − 5.92i)17-s + ⋯
L(s)  = 1  + (−0.998 + 0.0585i)2-s + (0.377 + 0.653i)3-s + (0.993 − 0.116i)4-s + (0.747 + 0.664i)5-s + (−0.414 − 0.630i)6-s + (0.391 + 0.920i)7-s + (−0.984 + 0.174i)8-s + (0.215 − 0.373i)9-s + (−0.785 − 0.619i)10-s + (0.0508 + 0.0880i)11-s + (0.451 + 0.604i)12-s + 0.360i·13-s + (−0.444 − 0.895i)14-s + (−0.151 + 0.739i)15-s + (0.972 − 0.232i)16-s + (−0.829 − 1.43i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.271 - 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.271 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.271 - 0.962i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.271 - 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.873673 + 0.661486i\)
\(L(\frac12)\) \(\approx\) \(0.873673 + 0.661486i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 - 0.0828i)T \)
5 \( 1 + (-1.67 - 1.48i)T \)
7 \( 1 + (-1.03 - 2.43i)T \)
good3 \( 1 + (-0.653 - 1.13i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (-0.168 - 0.292i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.29iT - 13T^{2} \)
17 \( 1 + (3.41 + 5.92i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.551 + 0.318i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.51 - 2.62i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 5.90iT - 29T^{2} \)
31 \( 1 + (-1.08 - 1.87i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.30 + 7.45i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 5.28iT - 41T^{2} \)
43 \( 1 - 10.6iT - 43T^{2} \)
47 \( 1 + (7.78 + 4.49i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.63 + 6.29i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7.50 + 4.33i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.97 + 6.87i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.9 + 6.31i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.53iT - 71T^{2} \)
73 \( 1 + (5.93 + 10.2i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.96 + 5.17i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.76T + 83T^{2} \)
89 \( 1 + (6.43 + 3.71i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 6.03T + 97T^{2} \)
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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

−11.62258019437655212306186200505, −10.99147979797695983126547321322, −9.787571319731425661492529349898, −9.375661156105185226292910324201, −8.580784130065444704556304315723, −7.18977062726625147584757534148, −6.34556298638009497860943963323, −5.07113180849998706462683456015, −3.17324034179639464765962851166, −2.03846678631141880671358076311, 1.23863607980736785881257926949, 2.32540635586293727186253095241, 4.33806939064703517525410134812, 6.00507042052476845859389978539, 6.95767402354661800585882155811, 8.165354574686925312596376513313, 8.436712598134793976031386381131, 9.889493434113271176306936932542, 10.43507211613104122647032822668, 11.48566864078582918390953058410

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