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Properties

Label 2-280-280.107-c1-0-2
Degree $2$
Conductor $280$
Sign $0.567 - 0.823i$
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  + (0.483 − 1.32i)2-s + (−3.11 − 0.835i)3-s + (−1.53 − 1.28i)4-s + (−1.40 + 1.74i)5-s + (−2.61 + 3.73i)6-s + (0.0485 − 2.64i)7-s + (−2.44 + 1.41i)8-s + (6.42 + 3.70i)9-s + (1.63 + 2.70i)10-s + (1.23 + 2.14i)11-s + (3.70 + 5.28i)12-s + (−0.837 − 0.837i)13-s + (−3.49 − 1.34i)14-s + (5.83 − 4.25i)15-s + (0.695 + 3.93i)16-s + (−1.64 + 6.14i)17-s + ⋯
L(s)  = 1  + (0.341 − 0.939i)2-s + (−1.80 − 0.482i)3-s + (−0.766 − 0.642i)4-s + (−0.628 + 0.778i)5-s + (−1.06 + 1.52i)6-s + (0.0183 − 0.999i)7-s + (−0.865 + 0.500i)8-s + (2.14 + 1.23i)9-s + (0.516 + 0.856i)10-s + (0.373 + 0.646i)11-s + (1.06 + 1.52i)12-s + (−0.232 − 0.232i)13-s + (−0.933 − 0.359i)14-s + (1.50 − 1.09i)15-s + (0.173 + 0.984i)16-s + (−0.399 + 1.49i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.567 - 0.823i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.567 - 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.567 - 0.823i$
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: \((2,\ 280,\ (\ :1/2),\ 0.567 - 0.823i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.197887 + 0.103904i\)
\(L(\frac12)\) \(\approx\) \(0.197887 + 0.103904i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.483 + 1.32i)T \)
5 \( 1 + (1.40 - 1.74i)T \)
7 \( 1 + (-0.0485 + 2.64i)T \)
good3 \( 1 + (3.11 + 0.835i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (-1.23 - 2.14i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.837 + 0.837i)T + 13iT^{2} \)
17 \( 1 + (1.64 - 6.14i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.588 - 0.339i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.50 - 0.670i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 3.80T + 29T^{2} \)
31 \( 1 + (4.91 - 2.83i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.04 - 3.91i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 0.957T + 41T^{2} \)
43 \( 1 + (4.19 - 4.19i)T - 43iT^{2} \)
47 \( 1 + (-0.232 - 0.868i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.712 - 2.66i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (8.10 - 4.67i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.10 - 0.639i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.72 + 6.43i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 0.973iT - 71T^{2} \)
73 \( 1 + (-1.64 - 0.439i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (7.01 - 12.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-10.2 + 10.2i)T - 83iT^{2} \)
89 \( 1 + (-3.10 - 1.79i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.33 + 1.33i)T + 97iT^{2} \)
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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

−11.92092968775767911353203191941, −11.03219704069424320021666132531, −10.63715931248205520282656339447, −9.850361773088430181483906876278, −7.84855529064748803161342148958, −6.83428984429387493105432690511, −5.99348764515694355132872857625, −4.67055598820698529691620234944, −3.81818837414471174672207301930, −1.58208635927413502583841091413, 0.19699205290875401483887287601, 3.87567030892055028886755363494, 4.96208688865490149482535568848, 5.49514596563232402659147030531, 6.46072526814586045441405365922, 7.54196851247775679060773221805, 8.956806424921127087480475965570, 9.556235488748952098585384116005, 11.23168212054745133031895896745, 11.80148402878835393308417284639

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