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Properties

Label 2-280-280.107-c1-0-25
Degree $2$
Conductor $280$
Sign $0.823 + 0.567i$
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  + (0.669 − 1.24i)2-s + (2.12 + 0.569i)3-s + (−1.10 − 1.66i)4-s + (0.177 + 2.22i)5-s + (2.13 − 2.26i)6-s + (2.48 + 0.895i)7-s + (−2.81 + 0.260i)8-s + (1.60 + 0.923i)9-s + (2.89 + 1.27i)10-s + (0.771 + 1.33i)11-s + (−1.39 − 4.17i)12-s + (−2.74 − 2.74i)13-s + (2.78 − 2.50i)14-s + (−0.892 + 4.84i)15-s + (−1.55 + 3.68i)16-s + (0.761 − 2.84i)17-s + ⋯
L(s)  = 1  + (0.473 − 0.881i)2-s + (1.22 + 0.329i)3-s + (−0.552 − 0.833i)4-s + (0.0793 + 0.996i)5-s + (0.870 − 0.926i)6-s + (0.940 + 0.338i)7-s + (−0.995 + 0.0922i)8-s + (0.533 + 0.307i)9-s + (0.915 + 0.401i)10-s + (0.232 + 0.402i)11-s + (−0.403 − 1.20i)12-s + (−0.760 − 0.760i)13-s + (0.743 − 0.668i)14-s + (−0.230 + 1.25i)15-s + (−0.389 + 0.920i)16-s + (0.184 − 0.689i)17-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(2.12214 - 0.660853i\)
\(L(\frac12)\) \(\approx\) \(2.12214 - 0.660853i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.669 + 1.24i)T \)
5 \( 1 + (-0.177 - 2.22i)T \)
7 \( 1 + (-2.48 - 0.895i)T \)
good3 \( 1 + (-2.12 - 0.569i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (-0.771 - 1.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.74 + 2.74i)T + 13iT^{2} \)
17 \( 1 + (-0.761 + 2.84i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (5.92 + 3.41i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.11 + 0.834i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 0.336T + 29T^{2} \)
31 \( 1 + (3.64 - 2.10i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.32 - 4.93i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 4.49T + 41T^{2} \)
43 \( 1 + (0.152 - 0.152i)T - 43iT^{2} \)
47 \( 1 + (2.69 + 10.0i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.43 - 5.35i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-4.09 + 2.36i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.10 - 1.79i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.77 - 14.0i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 9.29iT - 71T^{2} \)
73 \( 1 + (-14.3 - 3.85i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (3.18 - 5.50i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (11.6 - 11.6i)T - 83iT^{2} \)
89 \( 1 + (-6.46 - 3.72i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.88 + 3.88i)T + 97iT^{2} \)
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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

−11.65276648524289427430551272507, −10.83850610857813949997357760349, −9.967249722996502350605159087349, −9.101835854747679520915471005175, −8.165063694106176282236938906127, −6.91556593689494978124939725182, −5.33716075395712267731943714305, −4.20055056575571019054705436772, −2.91803613973020487626007572585, −2.23883592937603333590096692651, 1.98371945691092435387767130198, 3.80013322116805158677833453732, 4.67779589920060178565502680902, 5.93284065123792650440877089024, 7.34970381164687034267542117212, 8.112692571893801744531193525358, 8.706483930066379960968734595930, 9.504140616341657951757833344872, 11.21176232420235764840737639121, 12.45217860712631844895705478274

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