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Properties

Label 2-280-35.17-c1-0-11
Degree $2$
Conductor $280$
Sign $0.509 + 0.860i$
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  + (2.20 − 0.590i)3-s + (−0.352 − 2.20i)5-s + (−1.22 − 2.34i)7-s + (1.90 − 1.10i)9-s + (−0.0644 + 0.111i)11-s + (−0.748 − 0.748i)13-s + (−2.07 − 4.65i)15-s + (0.613 + 2.28i)17-s + (3.81 + 6.59i)19-s + (−4.08 − 4.44i)21-s + (4.11 + 1.10i)23-s + (−4.75 + 1.55i)25-s + (−1.28 + 1.28i)27-s − 0.163i·29-s + (8.43 + 4.86i)31-s + ⋯
L(s)  = 1  + (1.27 − 0.340i)3-s + (−0.157 − 0.987i)5-s + (−0.462 − 0.886i)7-s + (0.636 − 0.367i)9-s + (−0.0194 + 0.0336i)11-s + (−0.207 − 0.207i)13-s + (−0.536 − 1.20i)15-s + (0.148 + 0.555i)17-s + (0.874 + 1.51i)19-s + (−0.890 − 0.970i)21-s + (0.857 + 0.229i)23-s + (−0.950 + 0.310i)25-s + (−0.246 + 0.246i)27-s − 0.0303i·29-s + (1.51 + 0.874i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.509 + 0.860i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.509 + 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.509 + 0.860i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.509 + 0.860i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50860 - 0.860428i\)
\(L(\frac12)\) \(\approx\) \(1.50860 - 0.860428i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.352 + 2.20i)T \)
7 \( 1 + (1.22 + 2.34i)T \)
good3 \( 1 + (-2.20 + 0.590i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (0.0644 - 0.111i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.748 + 0.748i)T + 13iT^{2} \)
17 \( 1 + (-0.613 - 2.28i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-3.81 - 6.59i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.11 - 1.10i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 0.163iT - 29T^{2} \)
31 \( 1 + (-8.43 - 4.86i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.0241 + 0.0900i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 11.9iT - 41T^{2} \)
43 \( 1 + (-2.76 + 2.76i)T - 43iT^{2} \)
47 \( 1 + (-2.23 - 0.597i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.96 + 7.32i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (4.13 - 7.16i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (11.5 - 6.64i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.84 + 1.83i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 9.50T + 71T^{2} \)
73 \( 1 + (-1.80 + 0.483i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (8.48 - 4.89i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.43 + 8.43i)T + 83iT^{2} \)
89 \( 1 + (-3.37 - 5.85i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (9.61 - 9.61i)T - 97iT^{2} \)
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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

−12.05889150124973125141119846062, −10.51592764340469998122152728321, −9.642669948998939683269623902628, −8.716401097728909228941609675587, −7.932319550537232670681947351351, −7.16697705471609774208654741703, −5.60286669724159813508069081955, −4.14791103715188574035231093505, −3.15640726948108066072566763850, −1.38797649298996749552450063165, 2.71962460792157954001303201518, 3.01263511343626423871840887116, 4.63050878412770012056355237863, 6.20169658071746501631726612931, 7.26160784352457956359102690368, 8.253951283037114931640341171253, 9.347684081624242527928018104648, 9.718499988618507525310607328269, 11.11299320114832926035824214555, 11.88006716018259112296996140584

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