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Properties

Label 2-280-280.69-c2-0-57
Degree $2$
Conductor $280$
Sign $-0.474 + 0.880i$
Analytic cond. $7.62944$
Root an. cond. $2.76214$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.79 − 0.882i)2-s − 1.20i·3-s + (2.44 + 3.16i)4-s + (1.26 − 4.83i)5-s + (−1.06 + 2.15i)6-s + (−2.96 + 6.34i)7-s + (−1.58 − 7.84i)8-s + 7.55·9-s + (−6.54 + 7.56i)10-s − 6.35i·11-s + (3.80 − 2.93i)12-s − 11.4i·13-s + (10.9 − 8.76i)14-s + (−5.81 − 1.52i)15-s + (−4.08 + 15.4i)16-s + 17.8·17-s + ⋯
L(s)  = 1  + (−0.897 − 0.441i)2-s − 0.400i·3-s + (0.610 + 0.792i)4-s + (0.253 − 0.967i)5-s + (−0.176 + 0.359i)6-s + (−0.423 + 0.905i)7-s + (−0.197 − 0.980i)8-s + 0.839·9-s + (−0.654 + 0.756i)10-s − 0.577i·11-s + (0.317 − 0.244i)12-s − 0.881i·13-s + (0.779 − 0.625i)14-s + (−0.387 − 0.101i)15-s + (−0.255 + 0.966i)16-s + 1.05·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.474 + 0.880i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.474 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.474 + 0.880i$
Analytic conductor: \(7.62944\)
Root analytic conductor: \(2.76214\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1),\ -0.474 + 0.880i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.517532 - 0.866634i\)
\(L(\frac12)\) \(\approx\) \(0.517532 - 0.866634i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.79 + 0.882i)T \)
5 \( 1 + (-1.26 + 4.83i)T \)
7 \( 1 + (2.96 - 6.34i)T \)
good3 \( 1 + 1.20iT - 9T^{2} \)
11 \( 1 + 6.35iT - 121T^{2} \)
13 \( 1 + 11.4iT - 169T^{2} \)
17 \( 1 - 17.8T + 289T^{2} \)
19 \( 1 + 3.65T + 361T^{2} \)
23 \( 1 - 1.47iT - 529T^{2} \)
29 \( 1 + 13.9iT - 841T^{2} \)
31 \( 1 + 31.1iT - 961T^{2} \)
37 \( 1 + 10.8T + 1.36e3T^{2} \)
41 \( 1 + 60.7iT - 1.68e3T^{2} \)
43 \( 1 + 34.9T + 1.84e3T^{2} \)
47 \( 1 + 54.6T + 2.20e3T^{2} \)
53 \( 1 + 50.2T + 2.80e3T^{2} \)
59 \( 1 - 48.8T + 3.48e3T^{2} \)
61 \( 1 + 13.3T + 3.72e3T^{2} \)
67 \( 1 - 105.T + 4.48e3T^{2} \)
71 \( 1 - 65.2T + 5.04e3T^{2} \)
73 \( 1 - 97.4T + 5.32e3T^{2} \)
79 \( 1 + 12.8T + 6.24e3T^{2} \)
83 \( 1 - 140. iT - 6.88e3T^{2} \)
89 \( 1 - 25.4iT - 7.92e3T^{2} \)
97 \( 1 - 117.T + 9.40e3T^{2} \)
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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

−11.39313270811414599521201202968, −10.06730255162577699990864762327, −9.533956447385983983856055585541, −8.436857420733030312352256603777, −7.82982533594525132980412351971, −6.45325382547594157566857216526, −5.37533466493616758876861224877, −3.60220384661502342582789232724, −2.10751324869718802723183857007, −0.70931965810962849328311661728, 1.60962776014413376750360237309, 3.40673168916594222166811154361, 4.88229687449856364978267904927, 6.51443833989905390106825935857, 6.97703443949121055820352723063, 7.921802803639002010244278479598, 9.454597279075961638866924669096, 9.985244566490613467910570812989, 10.58769134012314818708422947092, 11.55521392723344002523031090098

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