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Properties

Label 2-280-35.17-c1-0-0
Degree $2$
Conductor $280$
Sign $-0.634 - 0.772i$
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  + (0.120 − 0.0322i)3-s + (−2.04 + 0.907i)5-s + (−2.46 + 0.969i)7-s + (−2.58 + 1.49i)9-s + (−1.40 + 2.44i)11-s + (3.28 + 3.28i)13-s + (−0.216 + 0.174i)15-s + (0.477 + 1.78i)17-s + (−4.01 − 6.95i)19-s + (−0.264 + 0.195i)21-s + (2.30 + 0.617i)23-s + (3.35 − 3.70i)25-s + (−0.526 + 0.526i)27-s + 8.63i·29-s + (−2.81 − 1.62i)31-s + ⋯
L(s)  = 1  + (0.0694 − 0.0186i)3-s + (−0.913 + 0.405i)5-s + (−0.930 + 0.366i)7-s + (−0.861 + 0.497i)9-s + (−0.424 + 0.736i)11-s + (0.911 + 0.911i)13-s + (−0.0559 + 0.0451i)15-s + (0.115 + 0.431i)17-s + (−0.921 − 1.59i)19-s + (−0.0577 + 0.0427i)21-s + (0.480 + 0.128i)23-s + (0.670 − 0.741i)25-s + (−0.101 + 0.101i)27-s + 1.60i·29-s + (−0.505 − 0.291i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.634 - 0.772i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.634 - 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.634 - 0.772i$
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: \((2,\ 280,\ (\ :1/2),\ -0.634 - 0.772i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.254780 + 0.539172i\)
\(L(\frac12)\) \(\approx\) \(0.254780 + 0.539172i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.04 - 0.907i)T \)
7 \( 1 + (2.46 - 0.969i)T \)
good3 \( 1 + (-0.120 + 0.0322i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (1.40 - 2.44i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.28 - 3.28i)T + 13iT^{2} \)
17 \( 1 + (-0.477 - 1.78i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (4.01 + 6.95i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.30 - 0.617i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 8.63iT - 29T^{2} \)
31 \( 1 + (2.81 + 1.62i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.87 - 6.99i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 9.45iT - 41T^{2} \)
43 \( 1 + (-1.04 + 1.04i)T - 43iT^{2} \)
47 \( 1 + (3.76 + 1.00i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.75 - 6.54i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (6.19 - 10.7i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.19 + 1.26i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-13.8 + 3.71i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 2.72T + 71T^{2} \)
73 \( 1 + (-3.90 + 1.04i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (5.52 - 3.18i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.41 - 7.41i)T + 83iT^{2} \)
89 \( 1 + (0.487 + 0.844i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.12 - 5.12i)T - 97iT^{2} \)
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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

−12.18274353951273124354857296941, −11.14904145117810373692124424827, −10.59570822400885470093316553938, −9.110444749217866026354249419581, −8.526255438013656088186653893069, −7.19858425407126622779982256383, −6.46655096538063211719325266765, −5.03547469423277431639997303431, −3.69984053404406222737383953529, −2.52940435723138468499960873826, 0.43213902321848154978711112508, 3.15298477867187221553437005717, 3.88561468371374309890664266063, 5.57875656156017942415218565949, 6.43363924023471392025509282133, 7.930831891807400015451157527918, 8.434304138254274263973629525651, 9.585350903721109398612242389775, 10.71807136888261878106649775503, 11.46964555857883708161724080261

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