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Properties

Label 2-280-280.237-c1-0-28
Degree $2$
Conductor $280$
Sign $0.847 + 0.530i$
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  + (−1.19 + 0.757i)2-s + (1.10 − 1.10i)3-s + (0.852 − 1.80i)4-s + (0.557 − 2.16i)5-s + (−0.480 + 2.14i)6-s + (1.45 + 2.20i)7-s + (0.353 + 2.80i)8-s + 0.577i·9-s + (0.974 + 3.00i)10-s − 2.81i·11-s + (−1.05 − 2.92i)12-s + (0.0608 − 0.0608i)13-s + (−3.41 − 1.52i)14-s + (−1.76 − 2.99i)15-s + (−2.54 − 3.08i)16-s + (4.42 − 4.42i)17-s + ⋯
L(s)  = 1  + (−0.844 + 0.535i)2-s + (0.635 − 0.635i)3-s + (0.426 − 0.904i)4-s + (0.249 − 0.968i)5-s + (−0.196 + 0.876i)6-s + (0.551 + 0.833i)7-s + (0.124 + 0.992i)8-s + 0.192i·9-s + (0.308 + 0.951i)10-s − 0.850i·11-s + (−0.304 − 0.845i)12-s + (0.0168 − 0.0168i)13-s + (−0.912 − 0.408i)14-s + (−0.456 − 0.773i)15-s + (−0.636 − 0.770i)16-s + (1.07 − 1.07i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.847 + 0.530i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.847 + 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.847 + 0.530i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.847 + 0.530i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11880 - 0.321426i\)
\(L(\frac12)\) \(\approx\) \(1.11880 - 0.321426i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.19 - 0.757i)T \)
5 \( 1 + (-0.557 + 2.16i)T \)
7 \( 1 + (-1.45 - 2.20i)T \)
good3 \( 1 + (-1.10 + 1.10i)T - 3iT^{2} \)
11 \( 1 + 2.81iT - 11T^{2} \)
13 \( 1 + (-0.0608 + 0.0608i)T - 13iT^{2} \)
17 \( 1 + (-4.42 + 4.42i)T - 17iT^{2} \)
19 \( 1 + 3.35iT - 19T^{2} \)
23 \( 1 + (0.363 - 0.363i)T - 23iT^{2} \)
29 \( 1 + 6.56T + 29T^{2} \)
31 \( 1 - 8.72iT - 31T^{2} \)
37 \( 1 + (-0.872 + 0.872i)T - 37iT^{2} \)
41 \( 1 + 11.4iT - 41T^{2} \)
43 \( 1 + (-5.71 - 5.71i)T + 43iT^{2} \)
47 \( 1 + (2.80 - 2.80i)T - 47iT^{2} \)
53 \( 1 + (-3.83 - 3.83i)T + 53iT^{2} \)
59 \( 1 - 3.60iT - 59T^{2} \)
61 \( 1 + 11.7T + 61T^{2} \)
67 \( 1 + (8.03 - 8.03i)T - 67iT^{2} \)
71 \( 1 - 10.1T + 71T^{2} \)
73 \( 1 + (-5.69 - 5.69i)T + 73iT^{2} \)
79 \( 1 - 6.61iT - 79T^{2} \)
83 \( 1 + (3.72 - 3.72i)T - 83iT^{2} \)
89 \( 1 + 6.89T + 89T^{2} \)
97 \( 1 + (-3.64 + 3.64i)T - 97iT^{2} \)
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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

−11.74168164758998300677866927564, −10.75385734590541812570330797445, −9.337654373860517829940629996210, −8.838492672904835591107078952335, −8.046229167123506652924089421533, −7.25476420790078505190760719124, −5.73227764998424411894524768904, −5.07349239219422434943370350522, −2.61861846714090063054870113950, −1.28199741039877545704730646913, 1.86480717144620228337294788863, 3.38177490339006732858527400135, 4.13582539355162739142238896971, 6.23029300717757305058828168165, 7.49168483322739802102973170029, 8.061197821977253425653777095054, 9.471412231520317089032190936644, 10.01138001139228669289191756866, 10.67486614486796388293999148814, 11.63303208952262316913503686178

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