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Properties

Label 2-280-280.107-c1-0-13
Degree $2$
Conductor $280$
Sign $0.553 - 0.832i$
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  + (1.38 + 0.289i)2-s + (−1.63 − 0.438i)3-s + (1.83 + 0.800i)4-s + (−1.55 + 1.60i)5-s + (−2.13 − 1.07i)6-s + (2.49 + 0.885i)7-s + (2.30 + 1.63i)8-s + (−0.114 − 0.0663i)9-s + (−2.61 + 1.77i)10-s + (2.94 + 5.09i)11-s + (−2.64 − 2.11i)12-s + (1.49 + 1.49i)13-s + (3.19 + 1.94i)14-s + (3.24 − 1.94i)15-s + (2.71 + 2.93i)16-s + (1.45 − 5.43i)17-s + ⋯
L(s)  = 1  + (0.978 + 0.204i)2-s + (−0.944 − 0.253i)3-s + (0.916 + 0.400i)4-s + (−0.694 + 0.719i)5-s + (−0.872 − 0.440i)6-s + (0.942 + 0.334i)7-s + (0.815 + 0.578i)8-s + (−0.0383 − 0.0221i)9-s + (−0.827 + 0.561i)10-s + (0.887 + 1.53i)11-s + (−0.764 − 0.609i)12-s + (0.414 + 0.414i)13-s + (0.854 + 0.520i)14-s + (0.838 − 0.503i)15-s + (0.679 + 0.733i)16-s + (0.353 − 1.31i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.553 - 0.832i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.553 - 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49058 + 0.799246i\)
\(L(\frac12)\) \(\approx\) \(1.49058 + 0.799246i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.38 - 0.289i)T \)
5 \( 1 + (1.55 - 1.60i)T \)
7 \( 1 + (-2.49 - 0.885i)T \)
good3 \( 1 + (1.63 + 0.438i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (-2.94 - 5.09i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.49 - 1.49i)T + 13iT^{2} \)
17 \( 1 + (-1.45 + 5.43i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (3.10 + 1.79i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.49 - 0.669i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 6.02T + 29T^{2} \)
31 \( 1 + (1.05 - 0.609i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.68 + 10.0i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 2.44T + 41T^{2} \)
43 \( 1 + (-7.70 + 7.70i)T - 43iT^{2} \)
47 \( 1 + (-1.11 - 4.15i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.25 + 4.68i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (2.51 - 1.45i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.34 + 3.66i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.21 + 8.27i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 2.68iT - 71T^{2} \)
73 \( 1 + (1.49 + 0.400i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-1.03 + 1.78i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.700 - 0.700i)T - 83iT^{2} \)
89 \( 1 + (-3.21 - 1.85i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.71 - 2.71i)T + 97iT^{2} \)
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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

−12.03675137592004754761643702443, −11.34737219031610165398907074340, −10.77183389443052210627962996001, −9.101974293614011455447980266592, −7.53424179491123642226908613965, −7.02342814061706368555167380909, −5.97498693472659608434961605414, −4.86385564460706433772281029319, −3.91128383119247512173579606067, −2.13697250509274190102788493021, 1.23628216571637748755477414315, 3.64774644588324970393322958194, 4.43852087177366905750920063167, 5.62383586177892799652026627765, 6.18008062217706818268361488715, 7.85390706258908586943567346010, 8.612693378693726024650068658725, 10.45539053614456930687008448968, 11.11183994533223146240943010114, 11.62570636343057826680456343773

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