Properties

Label 2-280-35.33-c1-0-10
Degree $2$
Conductor $280$
Sign $-0.0327 + 0.999i$
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  + (−0.0749 − 0.0200i)3-s + (−0.965 − 2.01i)5-s + (1.45 − 2.20i)7-s + (−2.59 − 1.49i)9-s + (−0.390 − 0.677i)11-s + (−3.28 + 3.28i)13-s + (0.0318 + 0.170i)15-s + (1.40 − 5.24i)17-s + (2.91 − 5.05i)19-s + (−0.153 + 0.136i)21-s + (8.39 − 2.24i)23-s + (−3.13 + 3.89i)25-s + (0.328 + 0.328i)27-s + 0.303i·29-s + (−5.04 + 2.91i)31-s + ⋯
L(s)  = 1  + (−0.0432 − 0.0115i)3-s + (−0.431 − 0.902i)5-s + (0.551 − 0.834i)7-s + (−0.864 − 0.498i)9-s + (−0.117 − 0.204i)11-s + (−0.911 + 0.911i)13-s + (0.00821 + 0.0440i)15-s + (0.341 − 1.27i)17-s + (0.669 − 1.15i)19-s + (−0.0335 + 0.0297i)21-s + (1.74 − 0.468i)23-s + (−0.627 + 0.778i)25-s + (0.0632 + 0.0632i)27-s + 0.0564i·29-s + (−0.905 + 0.522i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0327 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0327 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.0327 + 0.999i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.0327 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.723050 - 0.747137i\)
\(L(\frac12)\) \(\approx\) \(0.723050 - 0.747137i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.965 + 2.01i)T \)
7 \( 1 + (-1.45 + 2.20i)T \)
good3 \( 1 + (0.0749 + 0.0200i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (0.390 + 0.677i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.28 - 3.28i)T - 13iT^{2} \)
17 \( 1 + (-1.40 + 5.24i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-2.91 + 5.05i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-8.39 + 2.24i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 0.303iT - 29T^{2} \)
31 \( 1 + (5.04 - 2.91i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.90 - 7.12i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 4.39iT - 41T^{2} \)
43 \( 1 + (-7.33 - 7.33i)T + 43iT^{2} \)
47 \( 1 + (-2.28 + 0.611i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (1.61 - 6.01i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-2.56 - 4.43i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (8.91 + 5.14i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.47 + 0.395i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 5.83T + 71T^{2} \)
73 \( 1 + (-12.9 - 3.47i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-8.66 - 5.00i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.94 + 2.94i)T - 83iT^{2} \)
89 \( 1 + (-6.43 + 11.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.0900 + 0.0900i)T + 97iT^{2} \)
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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

−11.57251518332626550905094050238, −10.98850387698960394408166766746, −9.399375004380451222495334236439, −8.947173160640766918926667391537, −7.65165782470552871228090207815, −6.91461661138328310960773554369, −5.19919319704883261117946573713, −4.58217209792643988665786924109, −3.00684290827041592110931095750, −0.816685596772616207771010408214, 2.32743365224918834219475010208, 3.47658239030681718008113620780, 5.20259129510699119931968030093, 5.93244025162747008428653681273, 7.51786531136834404338917649942, 8.009491466307075728609976723575, 9.229018141450387582191357223824, 10.48021097212039843191446351433, 11.08076366151891606694438674119, 12.03872309892167943100121969830

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