Properties

Label 2-280-280.107-c1-0-22
Degree $2$
Conductor $280$
Sign $0.707 + 0.706i$
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  + (−0.499 − 1.32i)2-s + (3.11 + 0.833i)3-s + (−1.50 + 1.32i)4-s + (−1.44 − 1.70i)5-s + (−0.450 − 4.53i)6-s + (2.23 + 1.42i)7-s + (2.49 + 1.32i)8-s + (6.39 + 3.68i)9-s + (−1.53 + 2.76i)10-s + (−2.06 − 3.57i)11-s + (−5.77 + 2.85i)12-s + (1.73 + 1.73i)13-s + (0.770 − 3.66i)14-s + (−3.07 − 6.51i)15-s + (0.510 − 3.96i)16-s + (0.356 − 1.32i)17-s + ⋯
L(s)  = 1  + (−0.352 − 0.935i)2-s + (1.79 + 0.481i)3-s + (−0.750 + 0.660i)4-s + (−0.646 − 0.763i)5-s + (−0.183 − 1.85i)6-s + (0.842 + 0.538i)7-s + (0.882 + 0.469i)8-s + (2.13 + 1.22i)9-s + (−0.485 + 0.874i)10-s + (−0.622 − 1.07i)11-s + (−1.66 + 0.825i)12-s + (0.482 + 0.482i)13-s + (0.205 − 0.978i)14-s + (−0.793 − 1.68i)15-s + (0.127 − 0.991i)16-s + (0.0863 − 0.322i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.706i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.707 + 0.706i$
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: \((2,\ 280,\ (\ :1/2),\ 0.707 + 0.706i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57415 - 0.651037i\)
\(L(\frac12)\) \(\approx\) \(1.57415 - 0.651037i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.499 + 1.32i)T \)
5 \( 1 + (1.44 + 1.70i)T \)
7 \( 1 + (-2.23 - 1.42i)T \)
good3 \( 1 + (-3.11 - 0.833i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (2.06 + 3.57i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.73 - 1.73i)T + 13iT^{2} \)
17 \( 1 + (-0.356 + 1.32i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.51 + 0.872i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.71 - 0.994i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 0.469T + 29T^{2} \)
31 \( 1 + (-0.329 + 0.190i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.20 + 4.48i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 8.69T + 41T^{2} \)
43 \( 1 + (5.07 - 5.07i)T - 43iT^{2} \)
47 \( 1 + (0.284 + 1.06i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.00467 - 0.0174i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (7.49 - 4.32i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.35 + 3.66i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.65 + 9.92i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 11.5iT - 71T^{2} \)
73 \( 1 + (-7.60 - 2.03i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (3.90 - 6.76i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.47 + 2.47i)T - 83iT^{2} \)
89 \( 1 + (-6.00 - 3.46i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.49 + 8.49i)T + 97iT^{2} \)
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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

−11.63375537741730526081561669234, −10.76501196922541777871529328965, −9.593940253410357050373186058480, −8.741345885464355548733331301060, −8.347748125102652552085795681228, −7.68523916753716942502111262883, −5.03859547486689022523326056207, −4.05716365275011967470246613547, −3.07860686306565646249935472454, −1.77622048065076168495719598698, 1.87305660579739929382478615819, 3.55235172372003887197862894935, 4.60785474977520283305934962054, 6.59937861216653640700162313693, 7.47040811510517322722694052347, 7.997848127831512664612933485485, 8.591720982131685291529017773538, 9.970530924223154098106013754111, 10.53547306413900109312168643507, 12.25281787066508898316583969460

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