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Properties

Label 2-280-40.29-c1-0-12
Degree $2$
Conductor $280$
Sign $0.205 - 0.978i$
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 + 0.686i)2-s + 0.359·3-s + (1.05 + 1.69i)4-s + (−0.565 + 2.16i)5-s + (0.443 + 0.246i)6-s − i·7-s + (0.138 + 2.82i)8-s − 2.87·9-s + (−2.18 + 2.28i)10-s + 1.56i·11-s + (0.379 + 0.609i)12-s + 6.61·13-s + (0.686 − 1.23i)14-s + (−0.203 + 0.776i)15-s + (−1.76 + 3.58i)16-s − 2.81i·17-s + ⋯
L(s)  = 1  + (0.874 + 0.485i)2-s + 0.207·3-s + (0.528 + 0.849i)4-s + (−0.253 + 0.967i)5-s + (0.181 + 0.100i)6-s − 0.377i·7-s + (0.0490 + 0.998i)8-s − 0.957·9-s + (−0.691 + 0.722i)10-s + 0.472i·11-s + (0.109 + 0.176i)12-s + 1.83·13-s + (0.183 − 0.330i)14-s + (−0.0524 + 0.200i)15-s + (−0.442 + 0.896i)16-s − 0.683i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.205 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.205 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.205 - 0.978i$
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: \((2,\ 280,\ (\ :1/2),\ 0.205 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60692 + 1.30482i\)
\(L(\frac12)\) \(\approx\) \(1.60692 + 1.30482i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.23 - 0.686i)T \)
5 \( 1 + (0.565 - 2.16i)T \)
7 \( 1 + iT \)
good3 \( 1 - 0.359T + 3T^{2} \)
11 \( 1 - 1.56iT - 11T^{2} \)
13 \( 1 - 6.61T + 13T^{2} \)
17 \( 1 + 2.81iT - 17T^{2} \)
19 \( 1 + 5.37iT - 19T^{2} \)
23 \( 1 + 5.85iT - 23T^{2} \)
29 \( 1 - 6.75iT - 29T^{2} \)
31 \( 1 - 9.18T + 31T^{2} \)
37 \( 1 + 2.55T + 37T^{2} \)
41 \( 1 + 7.93T + 41T^{2} \)
43 \( 1 - 7.16T + 43T^{2} \)
47 \( 1 + 5.24iT - 47T^{2} \)
53 \( 1 + 7.46T + 53T^{2} \)
59 \( 1 + 3.87iT - 59T^{2} \)
61 \( 1 + 4.69iT - 61T^{2} \)
67 \( 1 + 9.92T + 67T^{2} \)
71 \( 1 - 11.4T + 71T^{2} \)
73 \( 1 + 3.67iT - 73T^{2} \)
79 \( 1 + 8.46T + 79T^{2} \)
83 \( 1 + 1.20T + 83T^{2} \)
89 \( 1 + 4.21T + 89T^{2} \)
97 \( 1 - 5.89iT - 97T^{2} \)
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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

−12.03415544980108418560571200790, −11.19895896261775555259045494434, −10.60374122966876841479736443330, −8.883951901276958311254242746320, −8.059410486277807393869222946047, −6.86057843297020969443454702068, −6.30427019005592438937274614578, −4.88958635843366943253204845095, −3.59701204780447989899542223363, −2.68736783842427175431550579576, 1.45464339030559807340903209319, 3.25262885752981937537056029733, 4.18676167574739688922664482974, 5.72285171525917174691560945514, 6.03792472714458055491643636285, 8.059923686746229294194852388203, 8.698605074744767558597993786265, 9.861175585627719232032707734482, 11.11014607408814480925824744165, 11.70906168338475581397897712249

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