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Properties

Label 2-280-280.213-c1-0-29
Degree $2$
Conductor $280$
Sign $0.844 + 0.534i$
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  + (−1.24 + 0.661i)2-s + (3.03 − 0.811i)3-s + (1.12 − 1.65i)4-s + (0.203 − 2.22i)5-s + (−3.24 + 3.01i)6-s + (−2.64 + 0.167i)7-s + (−0.310 + 2.81i)8-s + (5.92 − 3.42i)9-s + (1.21 + 2.91i)10-s + (1.83 + 1.06i)11-s + (2.06 − 5.92i)12-s + (0.498 + 0.498i)13-s + (3.18 − 1.95i)14-s + (−1.19 − 6.91i)15-s + (−1.47 − 3.71i)16-s + (−0.344 + 0.0923i)17-s + ⋯
L(s)  = 1  + (−0.883 + 0.467i)2-s + (1.74 − 0.468i)3-s + (0.562 − 0.827i)4-s + (0.0909 − 0.995i)5-s + (−1.32 + 1.23i)6-s + (−0.997 + 0.0633i)7-s + (−0.109 + 0.993i)8-s + (1.97 − 1.14i)9-s + (0.385 + 0.922i)10-s + (0.553 + 0.319i)11-s + (0.595 − 1.71i)12-s + (0.138 + 0.138i)13-s + (0.852 − 0.522i)14-s + (−0.307 − 1.78i)15-s + (−0.368 − 0.929i)16-s + (−0.0836 + 0.0224i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 + 0.534i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.844 + 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.844 + 0.534i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (213, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.844 + 0.534i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36799 - 0.396677i\)
\(L(\frac12)\) \(\approx\) \(1.36799 - 0.396677i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.24 - 0.661i)T \)
5 \( 1 + (-0.203 + 2.22i)T \)
7 \( 1 + (2.64 - 0.167i)T \)
good3 \( 1 + (-3.03 + 0.811i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (-1.83 - 1.06i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.498 - 0.498i)T + 13iT^{2} \)
17 \( 1 + (0.344 - 0.0923i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (3.42 - 1.97i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.24 + 4.65i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 3.32T + 29T^{2} \)
31 \( 1 + (-5.09 - 2.94i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.23 - 8.34i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 6.13iT - 41T^{2} \)
43 \( 1 + (6.70 - 6.70i)T - 43iT^{2} \)
47 \( 1 + (2.51 - 9.36i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (0.228 + 0.851i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-12.9 - 7.48i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.445 + 0.772i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.66 - 2.58i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 1.28T + 71T^{2} \)
73 \( 1 + (-0.278 - 1.04i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (11.4 - 6.61i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.33 - 5.33i)T + 83iT^{2} \)
89 \( 1 + (5.51 + 9.54i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.53 + 5.53i)T + 97iT^{2} \)
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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

−12.03535351003574720052042357696, −10.15608667875361903375183815544, −9.588529149498033808598908406049, −8.576674396858335122086610838288, −8.429426258085239995129580723327, −7.06969378844713329217932960145, −6.29582919574758428389049527247, −4.38626052047347215654428211996, −2.80096487093671239309304192233, −1.42634967454277669284347813903, 2.23209865130051588490233320298, 3.21181833893074481667912033320, 3.87021594755551237952719516913, 6.52369347501796286087020140681, 7.33315529851001179014544734832, 8.403922941778946393434146965487, 9.168871120751245273819149385207, 9.934069203047506268581275952864, 10.52933505765313374555218653010, 11.71392152102306007381865145725

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