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Properties

Label 2-280-280.107-c1-0-19
Degree $2$
Conductor $280$
Sign $0.847 - 0.531i$
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  + (−1.38 + 0.282i)2-s + (2.74 + 0.735i)3-s + (1.84 − 0.782i)4-s + (1.86 − 1.23i)5-s + (−4.00 − 0.244i)6-s + (−0.517 + 2.59i)7-s + (−2.33 + 1.60i)8-s + (4.38 + 2.53i)9-s + (−2.23 + 2.23i)10-s + (1.41 + 2.44i)11-s + (5.62 − 0.792i)12-s + (−4.50 − 4.50i)13-s + (−0.0146 − 3.74i)14-s + (6.02 − 2.01i)15-s + (2.77 − 2.87i)16-s + (0.0288 − 0.107i)17-s + ⋯
L(s)  = 1  + (−0.979 + 0.199i)2-s + (1.58 + 0.424i)3-s + (0.920 − 0.391i)4-s + (0.834 − 0.551i)5-s + (−1.63 − 0.0998i)6-s + (−0.195 + 0.980i)7-s + (−0.823 + 0.566i)8-s + (1.46 + 0.844i)9-s + (−0.707 + 0.707i)10-s + (0.425 + 0.737i)11-s + (1.62 − 0.228i)12-s + (−1.24 − 1.24i)13-s + (−0.00390 − 0.999i)14-s + (1.55 − 0.519i)15-s + (0.694 − 0.719i)16-s + (0.00700 − 0.0261i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.847 - 0.531i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.847 - 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.847 - 0.531i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.847 - 0.531i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43110 + 0.411408i\)
\(L(\frac12)\) \(\approx\) \(1.43110 + 0.411408i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.38 - 0.282i)T \)
5 \( 1 + (-1.86 + 1.23i)T \)
7 \( 1 + (0.517 - 2.59i)T \)
good3 \( 1 + (-2.74 - 0.735i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (-1.41 - 2.44i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.50 + 4.50i)T + 13iT^{2} \)
17 \( 1 + (-0.0288 + 0.107i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (3.58 + 2.07i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.50 - 1.20i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 5.85T + 29T^{2} \)
31 \( 1 + (-2.21 + 1.27i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.282 + 1.05i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 5.98T + 41T^{2} \)
43 \( 1 + (-3.01 + 3.01i)T - 43iT^{2} \)
47 \( 1 + (-2.16 - 8.09i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.53 - 5.73i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-4.70 + 2.71i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.201 - 0.116i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.423 - 1.58i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 0.772iT - 71T^{2} \)
73 \( 1 + (12.4 + 3.32i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-5.36 + 9.29i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.37 + 2.37i)T - 83iT^{2} \)
89 \( 1 + (7.48 + 4.32i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.14 - 3.14i)T + 97iT^{2} \)
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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

−12.09411321448771999001767087166, −10.25435896060643742123846701126, −9.834684867526526344717515647001, −9.049152911778937021324599544965, −8.428691188104660153841732190024, −7.50379501440393677613690179361, −6.11475902742923259070220394598, −4.80623471510313032286284744367, −2.80316675390644180450479229547, −2.08075237774716606692625290410, 1.73070325878522764606344380352, 2.73133776517396731509783108601, 3.93122456578579153658706366731, 6.51022065534334446926392369606, 7.02253000487274048178942257712, 8.095868764653406246047415067228, 8.917444072719660310317754036625, 9.841147894642329661781702760163, 10.31642733474246852370939123773, 11.67449872291196384875736707001

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