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Properties

Label 2-280-35.9-c1-0-9
Degree $2$
Conductor $280$
Sign $0.468 + 0.883i$
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  + (1.76 − 1.01i)3-s + (−1.80 − 1.31i)5-s + (2.27 − 1.34i)7-s + (0.577 − 1.00i)9-s + (−0.524 − 0.907i)11-s − 1.15i·13-s + (−4.53 − 0.489i)15-s + (5.90 − 3.40i)17-s + (−2.37 + 4.11i)19-s + (2.65 − 4.69i)21-s + (−2.76 − 1.59i)23-s + (1.51 + 4.76i)25-s + 3.76i·27-s − 5.14·29-s + (2.78 + 4.82i)31-s + ⋯
L(s)  = 1  + (1.01 − 0.588i)3-s + (−0.807 − 0.590i)5-s + (0.861 − 0.507i)7-s + (0.192 − 0.333i)9-s + (−0.158 − 0.273i)11-s − 0.320i·13-s + (−1.17 − 0.126i)15-s + (1.43 − 0.827i)17-s + (−0.544 + 0.943i)19-s + (0.579 − 1.02i)21-s + (−0.576 − 0.332i)23-s + (0.303 + 0.952i)25-s + 0.723i·27-s − 0.954·29-s + (0.500 + 0.866i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.468 + 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.468 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.468 + 0.883i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.468 + 0.883i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39447 - 0.838551i\)
\(L(\frac12)\) \(\approx\) \(1.39447 - 0.838551i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.80 + 1.31i)T \)
7 \( 1 + (-2.27 + 1.34i)T \)
good3 \( 1 + (-1.76 + 1.01i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (0.524 + 0.907i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.15iT - 13T^{2} \)
17 \( 1 + (-5.90 + 3.40i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.37 - 4.11i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.76 + 1.59i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.14T + 29T^{2} \)
31 \( 1 + (-2.78 - 4.82i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.25 - 2.45i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 9.68T + 41T^{2} \)
43 \( 1 - 7.44iT - 43T^{2} \)
47 \( 1 + (8.04 + 4.64i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.85 - 2.80i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.43 - 9.40i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.66 + 2.88i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.72 - 2.72i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.10T + 71T^{2} \)
73 \( 1 + (5.60 - 3.23i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.306 + 0.530i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.275iT - 83T^{2} \)
89 \( 1 + (8.63 - 14.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 7.59iT - 97T^{2} \)
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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

−11.83439550767974063360594555710, −10.85897249312259196703545074615, −9.651646150755942288122953499407, −8.364278937514627588108136461370, −8.002304774659482105726025037426, −7.28772639297321821156333299522, −5.52619538280165776785109778087, −4.27608452130594686944715188736, −3.05190067227242977402988168724, −1.34111996536675600625865716110, 2.33409470972099596783363440325, 3.57926296347906531110763768043, 4.50641732281407639718681657705, 6.01054923667603193679986980278, 7.57731924352626974157379582775, 8.123329839451813691139149769823, 9.095979077094337521270132478668, 10.05056245451364538944823726978, 11.11572169119995777766826088590, 11.85139500048949789532006679429

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