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Properties

Label 2-280-280.139-c1-0-1
Degree $2$
Conductor $280$
Sign $-0.247 - 0.968i$
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 + (0.206 + 3.73i)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 + (0.0552 + 0.998i)14-s + (0.294 − 0.398i)15-s + (0.0939 + 0.995i)16-s − 1.80·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.247 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.247 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.247 - 0.968i$
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: \((2,\ 280,\ (\ :1/2),\ -0.247 - 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.136496 + 0.175833i\)
\(L(\frac12)\) \(\approx\) \(0.136496 + 0.175833i\)
\(L(\frac{3}{2})\) 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

−11.78110871344792029334961098227, −11.38698565066964307897545226471, −10.65625801841092447054870955333, −9.299876502856174871033834907071, −8.795529674058179182757187385117, −6.77662231312149815639819807858, −6.26859068530167150489041468631, −4.74765005216250686214365107800, −3.58732825215689957858670592810, −2.44861953942632976857606953149, 0.15219327411975622829928244610, 3.46206398468902063259843652161, 4.44330130890383197220593386355, 5.66983034833472281425883256318, 6.47402851907952402786942762604, 7.56070103275236468892429240180, 8.667410363907050216921358084338, 9.297648313394971973421402814088, 10.75901823175975720312806677819, 11.89800749198030149122451411068

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