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Properties

Label 2-280-35.12-c1-0-5
Degree $2$
Conductor $280$
Sign $0.208 - 0.978i$
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  + (−0.617 + 2.30i)3-s + (2.21 + 0.313i)5-s + (2.53 − 0.755i)7-s + (−2.33 − 1.34i)9-s + (2.18 + 3.78i)11-s + (−4.36 − 4.36i)13-s + (−2.09 + 4.91i)15-s + (1.63 + 0.438i)17-s + (−3.56 + 6.17i)19-s + (0.175 + 6.31i)21-s + (−1.31 − 4.91i)23-s + (4.80 + 1.38i)25-s + (−0.510 + 0.510i)27-s − 1.33i·29-s + (−1.90 + 1.09i)31-s + ⋯
L(s)  = 1  + (−0.356 + 1.33i)3-s + (0.990 + 0.140i)5-s + (0.958 − 0.285i)7-s + (−0.778 − 0.449i)9-s + (0.659 + 1.14i)11-s + (−1.21 − 1.21i)13-s + (−0.539 + 1.26i)15-s + (0.396 + 0.106i)17-s + (−0.818 + 1.41i)19-s + (0.0382 + 1.37i)21-s + (−0.274 − 1.02i)23-s + (0.960 + 0.277i)25-s + (−0.0982 + 0.0982i)27-s − 0.247i·29-s + (−0.341 + 0.197i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.208 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.208 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.208 - 0.978i$
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: \((2,\ 280,\ (\ :1/2),\ 0.208 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09719 + 0.888188i\)
\(L(\frac12)\) \(\approx\) \(1.09719 + 0.888188i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.21 - 0.313i)T \)
7 \( 1 + (-2.53 + 0.755i)T \)
good3 \( 1 + (0.617 - 2.30i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (-2.18 - 3.78i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.36 + 4.36i)T + 13iT^{2} \)
17 \( 1 + (-1.63 - 0.438i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (3.56 - 6.17i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.31 + 4.91i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 1.33iT - 29T^{2} \)
31 \( 1 + (1.90 - 1.09i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.839 + 0.224i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 5.69iT - 41T^{2} \)
43 \( 1 + (-3.40 + 3.40i)T - 43iT^{2} \)
47 \( 1 + (2.63 + 9.84i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.02 - 0.541i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.56 + 4.44i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.21 - 3.58i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.90 + 7.11i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 4.31T + 71T^{2} \)
73 \( 1 + (3.82 - 14.2i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (4.82 + 2.78i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.272 - 0.272i)T + 83iT^{2} \)
89 \( 1 + (1.79 - 3.10i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.325 - 0.325i)T - 97iT^{2} \)
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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

−12.03683313494437793767143360207, −10.65387028870958109370915475519, −10.23522209354869461281075816819, −9.644931031205502763692797587812, −8.380305137407541533521521304459, −7.14182881566902134388187494852, −5.69441081149687435981211919063, −4.93701260267849132348448327051, −3.96374374007621555192726621443, −2.08844760263247506167943272565, 1.34217092321944856527523745390, 2.42200599362939655367420721893, 4.69458286907923153759158147466, 5.83999645430748326996546615430, 6.65708092432285761171193901917, 7.59740418068075153251768248605, 8.782968471866241873958543126030, 9.561125463727627498799415959496, 11.14605225941567210582168725292, 11.64453178571616904719825039768

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