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Properties

Label 2-280-35.24-c2-0-0
Degree $2$
Conductor $280$
Sign $-0.431 - 0.902i$
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  + (1.35 − 2.34i)3-s + (−4.98 + 0.425i)5-s + (−6.80 + 1.62i)7-s + (0.823 + 1.42i)9-s + (−4.96 + 8.59i)11-s + 4.98·13-s + (−5.75 + 12.2i)15-s + (−8.17 + 14.1i)17-s + (−27.7 + 16.0i)19-s + (−5.41 + 18.1i)21-s + (18.0 − 10.4i)23-s + (24.6 − 4.24i)25-s + 28.8·27-s + 2.05·29-s + (−39.4 − 22.7i)31-s + ⋯
L(s)  = 1  + (0.451 − 0.782i)3-s + (−0.996 + 0.0851i)5-s + (−0.972 + 0.231i)7-s + (0.0915 + 0.158i)9-s + (−0.450 + 0.781i)11-s + 0.383·13-s + (−0.383 + 0.818i)15-s + (−0.481 + 0.833i)17-s + (−1.46 + 0.843i)19-s + (−0.258 + 0.866i)21-s + (0.786 − 0.454i)23-s + (0.985 − 0.169i)25-s + 1.06·27-s + 0.0707·29-s + (−1.27 − 0.735i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.431 - 0.902i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.431 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.431 - 0.902i$
Analytic conductor: \(7.62944\)
Root analytic conductor: \(2.76214\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1),\ -0.431 - 0.902i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.285184 + 0.452489i\)
\(L(\frac12)\) \(\approx\) \(0.285184 + 0.452489i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (4.98 - 0.425i)T \)
7 \( 1 + (6.80 - 1.62i)T \)
good3 \( 1 + (-1.35 + 2.34i)T + (-4.5 - 7.79i)T^{2} \)
11 \( 1 + (4.96 - 8.59i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 4.98T + 169T^{2} \)
17 \( 1 + (8.17 - 14.1i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (27.7 - 16.0i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-18.0 + 10.4i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 - 2.05T + 841T^{2} \)
31 \( 1 + (39.4 + 22.7i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (41.2 - 23.7i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 63.1iT - 1.68e3T^{2} \)
43 \( 1 - 7.62iT - 1.84e3T^{2} \)
47 \( 1 + (41.7 + 72.2i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (12.8 + 7.40i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (28.6 + 16.5i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (83.3 - 48.1i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (41.6 + 24.0i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 56.9T + 5.04e3T^{2} \)
73 \( 1 + (7.01 - 12.1i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-24.1 - 41.8i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 9.81T + 6.88e3T^{2} \)
89 \( 1 + (-67.5 + 38.9i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 27.7T + 9.40e3T^{2} \)
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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

−12.29991026470227527560984322659, −10.96175615573765443921352981495, −10.16734010277026834052021933015, −8.766486654777440654006137416566, −8.081093117979794671026448923398, −7.09274914258856723692596033532, −6.31285122052916510463155951835, −4.58772073962655674926693855191, −3.36603617693352543511723315367, −1.97998224324824481892293007812, 0.24212058453722975190455215164, 3.04104353654054328366255249810, 3.78680316068259195415565874914, 4.90268295729877896126380826212, 6.49301351882123359047573268641, 7.41862009367708908827355565117, 8.886176534094886436655722723213, 9.079508378137073256781022226118, 10.58985235835345871158926207484, 11.01450871659212998305905164091

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