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Properties

Label 2-280-280.107-c1-0-0
Degree $2$
Conductor $280$
Sign $-0.720 - 0.693i$
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  + (−1.17 − 0.786i)2-s + (−2.06 − 0.552i)3-s + (0.761 + 1.84i)4-s + (2.22 + 0.200i)5-s + (1.98 + 2.27i)6-s + (−2.64 − 0.126i)7-s + (0.559 − 2.77i)8-s + (1.34 + 0.775i)9-s + (−2.45 − 1.98i)10-s + (−0.610 − 1.05i)11-s + (−0.549 − 4.23i)12-s + (−2.68 − 2.68i)13-s + (3.00 + 2.22i)14-s + (−4.47 − 1.64i)15-s + (−2.83 + 2.81i)16-s + (−1.84 + 6.90i)17-s + ⋯
L(s)  = 1  + (−0.830 − 0.556i)2-s + (−1.18 − 0.318i)3-s + (0.380 + 0.924i)4-s + (0.995 + 0.0895i)5-s + (0.811 + 0.926i)6-s + (−0.998 − 0.0477i)7-s + (0.197 − 0.980i)8-s + (0.447 + 0.258i)9-s + (−0.777 − 0.628i)10-s + (−0.183 − 0.318i)11-s + (−0.158 − 1.22i)12-s + (−0.745 − 0.745i)13-s + (0.803 + 0.595i)14-s + (−1.15 − 0.424i)15-s + (−0.709 + 0.704i)16-s + (−0.448 + 1.67i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.720 - 0.693i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.720 - 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.720 - 0.693i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.720 - 0.693i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.000503952 + 0.00124940i\)
\(L(\frac12)\) \(\approx\) \(0.000503952 + 0.00124940i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.17 + 0.786i)T \)
5 \( 1 + (-2.22 - 0.200i)T \)
7 \( 1 + (2.64 + 0.126i)T \)
good3 \( 1 + (2.06 + 0.552i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (0.610 + 1.05i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.68 + 2.68i)T + 13iT^{2} \)
17 \( 1 + (1.84 - 6.90i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.485 + 0.280i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.88 - 1.57i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 6.36T + 29T^{2} \)
31 \( 1 + (5.87 - 3.38i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.15 + 4.32i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 11.6T + 41T^{2} \)
43 \( 1 + (-3.61 + 3.61i)T - 43iT^{2} \)
47 \( 1 + (-0.551 - 2.05i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.01 + 3.77i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-2.73 + 1.58i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.85 + 2.22i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.392 - 1.46i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 3.28iT - 71T^{2} \)
73 \( 1 + (-3.27 - 0.877i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-4.28 + 7.42i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (9.39 - 9.39i)T - 83iT^{2} \)
89 \( 1 + (-12.2 - 7.08i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.07 + 7.07i)T + 97iT^{2} \)
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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

−12.30066811187935187149513655883, −10.97820896286569662947622819309, −10.43796201453593184841911415077, −9.671678564646605711918977295609, −8.595553467703982779251313394793, −7.21492456907833176991878382862, −6.28214771299291536886406613684, −5.53871804536119341554672340343, −3.52828527355233547156488108055, −1.94100436354535473433795896207, 0.00140525551457095793779632606, 2.28087719538363801576861634332, 4.79763686693454209283767983193, 5.66275071823912301303303801824, 6.49565057069518805451955012850, 7.27928591591763269621683030095, 8.989871089698228156677660160490, 9.726336173112678317411089839733, 10.22915339058042600509144719440, 11.33264368276837446354893418869

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