Properties

Label 2-280-35.12-c1-0-1
Degree $2$
Conductor $280$
Sign $-0.969 - 0.243i$
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  + (−0.536 + 2.00i)3-s + (−0.829 + 2.07i)5-s + (−2.33 + 1.24i)7-s + (−1.11 − 0.644i)9-s + (−3.09 − 5.36i)11-s + (−0.782 − 0.782i)13-s + (−3.70 − 2.77i)15-s + (4.40 + 1.18i)17-s + (−2.37 + 4.11i)19-s + (−1.25 − 5.33i)21-s + (1.74 + 6.52i)23-s + (−3.62 − 3.44i)25-s + (−2.50 + 2.50i)27-s + 5.30i·29-s + (2.16 − 1.25i)31-s + ⋯
L(s)  = 1  + (−0.309 + 1.15i)3-s + (−0.370 + 0.928i)5-s + (−0.881 + 0.472i)7-s + (−0.372 − 0.214i)9-s + (−0.933 − 1.61i)11-s + (−0.217 − 0.217i)13-s + (−0.957 − 0.715i)15-s + (1.06 + 0.286i)17-s + (−0.544 + 0.943i)19-s + (−0.272 − 1.16i)21-s + (0.364 + 1.35i)23-s + (−0.724 − 0.688i)25-s + (−0.482 + 0.482i)27-s + 0.985i·29-s + (0.389 − 0.224i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 - 0.243i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.969 - 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.969 - 0.243i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.969 - 0.243i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0858820 + 0.693391i\)
\(L(\frac12)\) \(\approx\) \(0.0858820 + 0.693391i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.829 - 2.07i)T \)
7 \( 1 + (2.33 - 1.24i)T \)
good3 \( 1 + (0.536 - 2.00i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (3.09 + 5.36i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.782 + 0.782i)T + 13iT^{2} \)
17 \( 1 + (-4.40 - 1.18i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (2.37 - 4.11i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.74 - 6.52i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 5.30iT - 29T^{2} \)
31 \( 1 + (-2.16 + 1.25i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (6.61 - 1.77i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 2.51iT - 41T^{2} \)
43 \( 1 + (-0.404 + 0.404i)T - 43iT^{2} \)
47 \( 1 + (-2.31 - 8.63i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-9.92 - 2.66i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.0710 + 0.123i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.67 - 1.54i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.16 + 4.34i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 + (0.483 - 1.80i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (5.42 + 3.13i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.82 + 5.82i)T + 83iT^{2} \)
89 \( 1 + (-4.58 + 7.94i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.64 - 2.64i)T - 97iT^{2} \)
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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

−12.09125918213367932487653921435, −11.06159961480159904302572052814, −10.42755036635804574116448954525, −9.788392744115492077761311227632, −8.555220437322587150264020448588, −7.47958038414813851187691031574, −6.03520127200216721824994425294, −5.38999587559554500159179426933, −3.65625137439770895887089109385, −3.08097765921681060290281806757, 0.53528494989144854078230000363, 2.29832228432794155329773610556, 4.22473826890125687648764454750, 5.32951148130624708684518176780, 6.80844384556410357028633372173, 7.28818853148917752364292615738, 8.328529116273957143239266530520, 9.599905681587148729397656298970, 10.38881090928410506674387933540, 11.89953450339838216925898081990

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