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Properties

Label 2-280-280.139-c1-0-13
Degree $2$
Conductor $280$
Sign $-0.563 - 0.826i$
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  + (0.510 + 1.31i)2-s + 0.857·3-s + (−1.47 + 1.34i)4-s + (1.33 + 1.79i)5-s + (0.437 + 1.13i)6-s + (−2.41 + 1.08i)7-s + (−2.53 − 1.26i)8-s − 2.26·9-s + (−1.69 + 2.67i)10-s + 3.05·11-s + (−1.26 + 1.15i)12-s + 3.18i·13-s + (−2.66 − 2.62i)14-s + (1.14 + 1.54i)15-s + (0.375 − 3.98i)16-s + 7.44·17-s + ⋯
L(s)  = 1  + (0.360 + 0.932i)2-s + 0.495·3-s + (−0.739 + 0.673i)4-s + (0.594 + 0.803i)5-s + (0.178 + 0.461i)6-s + (−0.911 + 0.411i)7-s + (−0.894 − 0.446i)8-s − 0.754·9-s + (−0.534 + 0.844i)10-s + 0.921·11-s + (−0.366 + 0.333i)12-s + 0.883i·13-s + (−0.712 − 0.701i)14-s + (0.294 + 0.398i)15-s + (0.0939 − 0.995i)16-s + 1.80·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.563 - 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.563 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.563 - 0.826i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.563 - 0.826i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.746413 + 1.41218i\)
\(L(\frac12)\) \(\approx\) \(0.746413 + 1.41218i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.510 - 1.31i)T \)
5 \( 1 + (-1.33 - 1.79i)T \)
7 \( 1 + (2.41 - 1.08i)T \)
good3 \( 1 - 0.857T + 3T^{2} \)
11 \( 1 - 3.05T + 11T^{2} \)
13 \( 1 - 3.18iT - 13T^{2} \)
17 \( 1 - 7.44T + 17T^{2} \)
19 \( 1 + 4.61iT - 19T^{2} \)
23 \( 1 - 0.708T + 23T^{2} \)
29 \( 1 + 2.41iT - 29T^{2} \)
31 \( 1 - 5.14T + 31T^{2} \)
37 \( 1 + 2.07T + 37T^{2} \)
41 \( 1 - 5.51iT - 41T^{2} \)
43 \( 1 + 4.21iT - 43T^{2} \)
47 \( 1 + 6.55iT - 47T^{2} \)
53 \( 1 + 3.75T + 53T^{2} \)
59 \( 1 + 3.11iT - 59T^{2} \)
61 \( 1 - 9.55T + 61T^{2} \)
67 \( 1 + 0.519iT - 67T^{2} \)
71 \( 1 + 10.9iT - 71T^{2} \)
73 \( 1 - 2.32T + 73T^{2} \)
79 \( 1 - 9.98iT - 79T^{2} \)
83 \( 1 + 8.73T + 83T^{2} \)
89 \( 1 - 18.5iT - 89T^{2} \)
97 \( 1 + 1.58T + 97T^{2} \)
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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

−12.29764891834482973051727821868, −11.48472872122428877553414336658, −9.808595764865570620603579912169, −9.318028584178748377302215278623, −8.317643724290248028526690173748, −7.01562525846962262406033990944, −6.35132375865902190056444165885, −5.41694129111998011443457481579, −3.68442501364163505221260604156, −2.77531108068915748050555163969, 1.16276053621444108017457375403, 2.94443404859985152340020576163, 3.84503945904065494934005181356, 5.40031920659103768170261509614, 6.13301725647189470537444885466, 7.992162823140482574770623140749, 8.959264140932360398468710950351, 9.789397623815153282746062411889, 10.35894722280537123486229819527, 11.81146727789500099503772032028

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