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Properties

Label 2-280-35.4-c1-0-2
Degree $2$
Conductor $280$
Sign $0.436 - 0.899i$
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  + (−1.82 − 1.05i)3-s + (1.46 + 1.68i)5-s + (−1.44 + 2.21i)7-s + (0.722 + 1.25i)9-s + (−0.887 + 1.53i)11-s + 1.44i·13-s + (−0.898 − 4.62i)15-s + (5.08 + 2.93i)17-s + (3.58 + 6.20i)19-s + (4.97 − 2.51i)21-s + (0.574 − 0.331i)23-s + (−0.698 + 4.95i)25-s + 3.27i·27-s − 6.45·29-s + (5.03 − 8.72i)31-s + ⋯
L(s)  = 1  + (−1.05 − 0.608i)3-s + (0.655 + 0.754i)5-s + (−0.547 + 0.836i)7-s + (0.240 + 0.417i)9-s + (−0.267 + 0.463i)11-s + 0.400i·13-s + (−0.231 − 1.19i)15-s + (1.23 + 0.711i)17-s + (0.821 + 1.42i)19-s + (1.08 − 0.548i)21-s + (0.119 − 0.0691i)23-s + (−0.139 + 0.990i)25-s + 0.631i·27-s − 1.19·29-s + (0.904 − 1.56i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.436 - 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.436 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.436 - 0.899i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.436 - 0.899i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.720081 + 0.451173i\)
\(L(\frac12)\) \(\approx\) \(0.720081 + 0.451173i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.46 - 1.68i)T \)
7 \( 1 + (1.44 - 2.21i)T \)
good3 \( 1 + (1.82 + 1.05i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (0.887 - 1.53i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.44iT - 13T^{2} \)
17 \( 1 + (-5.08 - 2.93i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.58 - 6.20i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.574 + 0.331i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.45T + 29T^{2} \)
31 \( 1 + (-5.03 + 8.72i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.34 - 2.50i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 1.92T + 41T^{2} \)
43 \( 1 + 5.81iT - 43T^{2} \)
47 \( 1 + (3.20 - 1.85i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.513 + 0.296i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.79 - 6.57i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.36 + 7.55i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.87 - 1.08i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.49T + 71T^{2} \)
73 \( 1 + (-13.4 - 7.76i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.35 + 2.34i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.35iT - 83T^{2} \)
89 \( 1 + (-2.93 - 5.07i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 14.7iT - 97T^{2} \)
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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

−12.10818517599798943771789936340, −11.24594736109828969640124916574, −10.11594547666225826101259916876, −9.516668605907098781774024241633, −7.937141773706510257116252864970, −6.86905636386195558917198173723, −5.93606909312407444430920639972, −5.49886846491221106706953869377, −3.41243063697954742659969070885, −1.82863353508091121116997731933, 0.75844383396825518542747489938, 3.24037210440897876615507653616, 4.86274497007791068608013183645, 5.37640578906352351354038548052, 6.51222777335645832136301141684, 7.74166067637623630907706805651, 9.153496450328187742686693705147, 9.949344246622962023593815670933, 10.65986548179725913994717189275, 11.58675879403482149678870954793

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