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Properties

Label 2-280-280.69-c2-0-47
Degree $2$
Conductor $280$
Sign $0.939 + 0.342i$
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  + (−1.65 − 1.12i)2-s + 2.74i·3-s + (1.47 + 3.71i)4-s + (4.72 + 1.63i)5-s + (3.09 − 4.54i)6-s + (−1.41 − 6.85i)7-s + (1.75 − 7.80i)8-s + 1.43·9-s + (−5.98 − 8.01i)10-s − 14.1i·11-s + (−10.2 + 4.04i)12-s − 5.89i·13-s + (−5.37 + 12.9i)14-s + (−4.48 + 12.9i)15-s + (−11.6 + 10.9i)16-s + 10.0·17-s + ⋯
L(s)  = 1  + (−0.826 − 0.562i)2-s + 0.916i·3-s + (0.367 + 0.929i)4-s + (0.945 + 0.326i)5-s + (0.515 − 0.757i)6-s + (−0.201 − 0.979i)7-s + (0.218 − 0.975i)8-s + 0.159·9-s + (−0.598 − 0.801i)10-s − 1.29i·11-s + (−0.852 + 0.336i)12-s − 0.453i·13-s + (−0.384 + 0.923i)14-s + (−0.299 + 0.866i)15-s + (−0.729 + 0.683i)16-s + 0.588·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.939 + 0.342i$
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: \((2,\ 280,\ (\ :1),\ 0.939 + 0.342i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.30725 - 0.231185i\)
\(L(\frac12)\) \(\approx\) \(1.30725 - 0.231185i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.65 + 1.12i)T \)
5 \( 1 + (-4.72 - 1.63i)T \)
7 \( 1 + (1.41 + 6.85i)T \)
good3 \( 1 - 2.74iT - 9T^{2} \)
11 \( 1 + 14.1iT - 121T^{2} \)
13 \( 1 + 5.89iT - 169T^{2} \)
17 \( 1 - 10.0T + 289T^{2} \)
19 \( 1 - 18.9T + 361T^{2} \)
23 \( 1 + 11.5iT - 529T^{2} \)
29 \( 1 - 31.7iT - 841T^{2} \)
31 \( 1 + 48.2iT - 961T^{2} \)
37 \( 1 + 39.7T + 1.36e3T^{2} \)
41 \( 1 - 15.8iT - 1.68e3T^{2} \)
43 \( 1 - 30.0T + 1.84e3T^{2} \)
47 \( 1 - 83.3T + 2.20e3T^{2} \)
53 \( 1 - 17.9T + 2.80e3T^{2} \)
59 \( 1 + 109.T + 3.48e3T^{2} \)
61 \( 1 - 35.4T + 3.72e3T^{2} \)
67 \( 1 - 44.5T + 4.48e3T^{2} \)
71 \( 1 + 75.5T + 5.04e3T^{2} \)
73 \( 1 - 81.9T + 5.32e3T^{2} \)
79 \( 1 - 89.6T + 6.24e3T^{2} \)
83 \( 1 - 107. iT - 6.88e3T^{2} \)
89 \( 1 - 145. iT - 7.92e3T^{2} \)
97 \( 1 - 13.4T + 9.40e3T^{2} \)
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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

−10.88950672439739439516929317330, −10.68950009190682106840972778542, −9.766735717169770940856875886582, −9.168631836472747841977709447892, −7.87461420027393415552856102531, −6.82071005107340379379787744613, −5.49239133564295085646380002619, −3.88861950351178499883323895102, −2.98348191687022968554454759794, −1.03807668569967865307768891082, 1.39626770624733496662490963784, 2.30645233885298253613287311689, 5.00964003361756240145644537905, 5.94108212321689422363840738912, 6.90051945479807492225972700960, 7.66891049452705007777464073262, 8.919500414363064461976969208435, 9.582484571076311851597176084244, 10.35320412387448817726228564841, 11.98206022567671837322250290412

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