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Properties

Label 2-280-40.27-c1-0-26
Degree $2$
Conductor $280$
Sign $0.945 + 0.324i$
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  + (1.39 + 0.212i)2-s + (0.570 − 0.570i)3-s + (1.90 + 0.595i)4-s + (−0.776 − 2.09i)5-s + (0.918 − 0.675i)6-s + (0.707 − 0.707i)7-s + (2.54 + 1.23i)8-s + 2.35i·9-s + (−0.639 − 3.09i)10-s − 1.03·11-s + (1.42 − 0.749i)12-s + (−1.92 − 1.92i)13-s + (1.13 − 0.838i)14-s + (−1.63 − 0.752i)15-s + (3.29 + 2.27i)16-s + (0.113 + 0.113i)17-s + ⋯
L(s)  = 1  + (0.988 + 0.150i)2-s + (0.329 − 0.329i)3-s + (0.954 + 0.297i)4-s + (−0.347 − 0.937i)5-s + (0.374 − 0.275i)6-s + (0.267 − 0.267i)7-s + (0.899 + 0.437i)8-s + 0.783i·9-s + (−0.202 − 0.979i)10-s − 0.312·11-s + (0.412 − 0.216i)12-s + (−0.534 − 0.534i)13-s + (0.304 − 0.224i)14-s + (−0.422 − 0.194i)15-s + (0.822 + 0.568i)16-s + (0.0275 + 0.0275i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.324i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 + 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.945 + 0.324i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.945 + 0.324i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.34839 - 0.391269i\)
\(L(\frac12)\) \(\approx\) \(2.34839 - 0.391269i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 - 0.212i)T \)
5 \( 1 + (0.776 + 2.09i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (-0.570 + 0.570i)T - 3iT^{2} \)
11 \( 1 + 1.03T + 11T^{2} \)
13 \( 1 + (1.92 + 1.92i)T + 13iT^{2} \)
17 \( 1 + (-0.113 - 0.113i)T + 17iT^{2} \)
19 \( 1 - 0.879iT - 19T^{2} \)
23 \( 1 + (-0.250 - 0.250i)T + 23iT^{2} \)
29 \( 1 + 2.57T + 29T^{2} \)
31 \( 1 - 5.78iT - 31T^{2} \)
37 \( 1 + (6.90 - 6.90i)T - 37iT^{2} \)
41 \( 1 + 7.82T + 41T^{2} \)
43 \( 1 + (-7.89 + 7.89i)T - 43iT^{2} \)
47 \( 1 + (0.137 - 0.137i)T - 47iT^{2} \)
53 \( 1 + (6.52 + 6.52i)T + 53iT^{2} \)
59 \( 1 + 11.9iT - 59T^{2} \)
61 \( 1 - 1.41iT - 61T^{2} \)
67 \( 1 + (-6.46 - 6.46i)T + 67iT^{2} \)
71 \( 1 + 3.26iT - 71T^{2} \)
73 \( 1 + (-3.50 + 3.50i)T - 73iT^{2} \)
79 \( 1 + 1.49T + 79T^{2} \)
83 \( 1 + (-4.86 + 4.86i)T - 83iT^{2} \)
89 \( 1 - 12.9iT - 89T^{2} \)
97 \( 1 + (7.01 + 7.01i)T + 97iT^{2} \)
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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

−12.15687635194720876789365415876, −11.09676776193106248073459531923, −10.14552458934747363743234021866, −8.522212861767103321984521803988, −7.85646713787000583592074699788, −6.97717419190997160775084961176, −5.36686471986504215447289433032, −4.80507457443025244831722949717, −3.40509054307973405894968414722, −1.85182180253547418552840612871, 2.37219363825570380823703391584, 3.47122680460536844910083689377, 4.47538656928868791970744532595, 5.85031722375095780815803818915, 6.85766686409706668607811226062, 7.75524198868816202613071055996, 9.239426650718928497453702096268, 10.24159023038689840346826786304, 11.17814397335669658921309365797, 11.89657167071457698247856524629

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