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Properties

Label 2-280-280.67-c1-0-9
Degree $2$
Conductor $280$
Sign $-0.829 - 0.557i$
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  + (−0.624 + 1.26i)2-s + (0.629 + 2.34i)3-s + (−1.21 − 1.58i)4-s + (1.82 − 1.29i)5-s + (−3.37 − 0.668i)6-s + (0.511 + 2.59i)7-s + (2.77 − 0.557i)8-s + (−2.52 + 1.45i)9-s + (0.509 + 3.12i)10-s + (−2.68 + 4.65i)11-s + (2.95 − 3.86i)12-s + (−0.646 − 0.646i)13-s + (−3.61 − 0.972i)14-s + (4.19 + 3.45i)15-s + (−1.02 + 3.86i)16-s + (0.0643 − 0.0172i)17-s + ⋯
L(s)  = 1  + (−0.441 + 0.897i)2-s + (0.363 + 1.35i)3-s + (−0.609 − 0.792i)4-s + (0.814 − 0.580i)5-s + (−1.37 − 0.273i)6-s + (0.193 + 0.981i)7-s + (0.980 − 0.197i)8-s + (−0.841 + 0.485i)9-s + (0.161 + 0.986i)10-s + (−0.810 + 1.40i)11-s + (0.853 − 1.11i)12-s + (−0.179 − 0.179i)13-s + (−0.965 − 0.259i)14-s + (1.08 + 0.893i)15-s + (−0.256 + 0.966i)16-s + (0.0156 − 0.00418i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.829 - 0.557i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.829 - 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.829 - 0.557i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.829 - 0.557i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.346113 + 1.13513i\)
\(L(\frac12)\) \(\approx\) \(0.346113 + 1.13513i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.624 - 1.26i)T \)
5 \( 1 + (-1.82 + 1.29i)T \)
7 \( 1 + (-0.511 - 2.59i)T \)
good3 \( 1 + (-0.629 - 2.34i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (2.68 - 4.65i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.646 + 0.646i)T + 13iT^{2} \)
17 \( 1 + (-0.0643 + 0.0172i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.477 + 0.275i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.42 + 5.32i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 2.78T + 29T^{2} \)
31 \( 1 + (8.21 + 4.74i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.71 - 1.53i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 4.40T + 41T^{2} \)
43 \( 1 + (2.49 - 2.49i)T - 43iT^{2} \)
47 \( 1 + (-8.32 - 2.23i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.798 - 0.213i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-5.94 - 3.43i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.60 - 2.08i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.00 + 1.87i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 12.4iT - 71T^{2} \)
73 \( 1 + (-1.06 - 3.97i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (7.94 + 13.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-9.13 + 9.13i)T - 83iT^{2} \)
89 \( 1 + (-12.2 + 7.05i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.04 + 6.04i)T + 97iT^{2} \)
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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

−12.44271240515870363972502742503, −10.74533048414041670580015060876, −9.972774958263976788060876291729, −9.344445944420477301192909135659, −8.720903549250690770367024370156, −7.59883086938090714203965365347, −6.06059694688692323281351843001, −5.06124619870607639173056226143, −4.52328209111237783932652961344, −2.32898722279192315409998494269, 1.10064816293271269327168198247, 2.40300699778332051153851477982, 3.50768432332000949206650677008, 5.48129260686297296046026458345, 6.92040272963846339904925353987, 7.62866750263369033945013304333, 8.534154335518185812736368800584, 9.683063197390222044012672154639, 10.74777165726054806304000336194, 11.25491176216439040203407100568

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