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Properties

Label 2-280-40.29-c1-0-34
Degree $2$
Conductor $280$
Sign $0.0163 + 0.999i$
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  + (0.614 − 1.27i)2-s + 2.96·3-s + (−1.24 − 1.56i)4-s + (−2.18 − 0.458i)5-s + (1.82 − 3.77i)6-s − i·7-s + (−2.75 + 0.625i)8-s + 5.80·9-s + (−1.92 + 2.50i)10-s − 0.338i·11-s + (−3.69 − 4.64i)12-s + 2.66·13-s + (−1.27 − 0.614i)14-s + (−6.49 − 1.36i)15-s + (−0.896 + 3.89i)16-s − 3.60i·17-s + ⋯
L(s)  = 1  + (0.434 − 0.900i)2-s + 1.71·3-s + (−0.622 − 0.782i)4-s + (−0.978 − 0.205i)5-s + (0.743 − 1.54i)6-s − 0.377i·7-s + (−0.975 + 0.221i)8-s + 1.93·9-s + (−0.609 + 0.792i)10-s − 0.102i·11-s + (−1.06 − 1.34i)12-s + 0.738·13-s + (−0.340 − 0.164i)14-s + (−1.67 − 0.351i)15-s + (−0.224 + 0.974i)16-s − 0.874i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0163 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0163 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.0163 + 0.999i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.0163 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53530 - 1.51038i\)
\(L(\frac12)\) \(\approx\) \(1.53530 - 1.51038i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.614 + 1.27i)T \)
5 \( 1 + (2.18 + 0.458i)T \)
7 \( 1 + iT \)
good3 \( 1 - 2.96T + 3T^{2} \)
11 \( 1 + 0.338iT - 11T^{2} \)
13 \( 1 - 2.66T + 13T^{2} \)
17 \( 1 + 3.60iT - 17T^{2} \)
19 \( 1 - 7.58iT - 19T^{2} \)
23 \( 1 - 1.51iT - 23T^{2} \)
29 \( 1 - 9.39iT - 29T^{2} \)
31 \( 1 - 3.04T + 31T^{2} \)
37 \( 1 + 5.70T + 37T^{2} \)
41 \( 1 + 6.41T + 41T^{2} \)
43 \( 1 + 7.73T + 43T^{2} \)
47 \( 1 + 10.6iT - 47T^{2} \)
53 \( 1 + 2.35T + 53T^{2} \)
59 \( 1 + 8.53iT - 59T^{2} \)
61 \( 1 - 2.39iT - 61T^{2} \)
67 \( 1 - 6.93T + 67T^{2} \)
71 \( 1 - 0.174T + 71T^{2} \)
73 \( 1 + 9.77iT - 73T^{2} \)
79 \( 1 - 12.4T + 79T^{2} \)
83 \( 1 + 4.33T + 83T^{2} \)
89 \( 1 + 15.1T + 89T^{2} \)
97 \( 1 - 7.09iT - 97T^{2} \)
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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

−11.79524836987829408871186367219, −10.62730892461773970925043473174, −9.752263020823215855803108110989, −8.685416579763394041176376985890, −8.173099882208365758817306627071, −6.94072399739433606361745464801, −4.99866458454154808277910681544, −3.65377265472513156403397802073, −3.36077263806379194722609938470, −1.60839938981265434724101132244, 2.72633250264245534632235714730, 3.71475873740972650277368274795, 4.62842027993694519690592032362, 6.44906356214400996992493447248, 7.39111528534228573291475383465, 8.376461856177610878641830933611, 8.626685404597968831548399357531, 9.783673363055208896295788543556, 11.32059175307252145206549720091, 12.47402895249943370946971781901

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