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Properties

Label 2-280-280.213-c1-0-21
Degree $2$
Conductor $280$
Sign $0.328 - 0.944i$
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  + (1.23 + 0.684i)2-s + (−0.533 + 0.142i)3-s + (1.06 + 1.69i)4-s + (1.41 + 1.73i)5-s + (−0.757 − 0.187i)6-s + (1.59 − 2.10i)7-s + (0.158 + 2.82i)8-s + (−2.33 + 1.34i)9-s + (0.566 + 3.11i)10-s + (−1.11 − 0.641i)11-s + (−0.809 − 0.750i)12-s + (−0.949 − 0.949i)13-s + (3.42 − 1.51i)14-s + (−1.00 − 0.721i)15-s + (−1.73 + 3.60i)16-s + (1.15 − 0.308i)17-s + ⋯
L(s)  = 1  + (0.875 + 0.483i)2-s + (−0.307 + 0.0824i)3-s + (0.531 + 0.846i)4-s + (0.632 + 0.774i)5-s + (−0.309 − 0.0767i)6-s + (0.604 − 0.796i)7-s + (0.0559 + 0.998i)8-s + (−0.778 + 0.449i)9-s + (0.179 + 0.983i)10-s + (−0.335 − 0.193i)11-s + (−0.233 − 0.216i)12-s + (−0.263 − 0.263i)13-s + (0.914 − 0.404i)14-s + (−0.258 − 0.186i)15-s + (−0.434 + 0.900i)16-s + (0.279 − 0.0748i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.328 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.328 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.328 - 0.944i$
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: \((2,\ 280,\ (\ :1/2),\ 0.328 - 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.66186 + 1.18102i\)
\(L(\frac12)\) \(\approx\) \(1.66186 + 1.18102i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.23 - 0.684i)T \)
5 \( 1 + (-1.41 - 1.73i)T \)
7 \( 1 + (-1.59 + 2.10i)T \)
good3 \( 1 + (0.533 - 0.142i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (1.11 + 0.641i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.949 + 0.949i)T + 13iT^{2} \)
17 \( 1 + (-1.15 + 0.308i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-6.26 + 3.61i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.710 - 2.65i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + 5.81T + 29T^{2} \)
31 \( 1 + (4.15 + 2.39i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.13 + 4.22i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 8.75iT - 41T^{2} \)
43 \( 1 + (-8.66 + 8.66i)T - 43iT^{2} \)
47 \( 1 + (-1.15 + 4.30i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.53 - 9.45i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (4.85 + 2.80i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.740 + 1.28i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.95 - 1.86i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 8.21T + 71T^{2} \)
73 \( 1 + (-2.84 - 10.6i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (1.55 - 0.899i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.32 - 6.32i)T + 83iT^{2} \)
89 \( 1 + (-1.07 - 1.86i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-10.7 - 10.7i)T + 97iT^{2} \)
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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

−11.98415964044925253268891295281, −11.09619455239946556299185607794, −10.60700136050505791667406192350, −9.152547089199685721418075692260, −7.62264808995947713542049032688, −7.24109087711093491936601133926, −5.70425852088552490414635954295, −5.28028078278161198182322043019, −3.70226387271847293832476236820, −2.43360842740878765309281191396, 1.54851645657392045292934322096, 2.96838277957128023966902525021, 4.66581677664419717725723877039, 5.53451148911435986394020623331, 6.12020496590482048637966133960, 7.76224595901175030462074779760, 9.081516757093249260080014135202, 9.816889995380204006898388166500, 11.07109655390298524490123315736, 11.91579020190920513889901270596

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