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Properties

Label 2-280-35.4-c1-0-7
Degree $2$
Conductor $280$
Sign $0.827 - 0.561i$
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  + (2.63 + 1.52i)3-s + (0.328 − 2.21i)5-s + (−0.629 + 2.56i)7-s + (3.12 + 5.42i)9-s + (−1.60 + 2.77i)11-s − 6.25i·13-s + (4.23 − 5.32i)15-s + (1.54 + 0.890i)17-s + (−1.83 − 3.18i)19-s + (−5.56 + 5.81i)21-s + (2.78 − 1.60i)23-s + (−4.78 − 1.45i)25-s + 9.92i·27-s − 2.88·29-s + (−1.40 + 2.44i)31-s + ⋯
L(s)  = 1  + (1.52 + 0.878i)3-s + (0.146 − 0.989i)5-s + (−0.237 + 0.971i)7-s + (1.04 + 1.80i)9-s + (−0.482 + 0.835i)11-s − 1.73i·13-s + (1.09 − 1.37i)15-s + (0.374 + 0.216i)17-s + (−0.421 − 0.730i)19-s + (−1.21 + 1.26i)21-s + (0.580 − 0.335i)23-s + (−0.956 − 0.290i)25-s + 1.90i·27-s − 0.535·29-s + (−0.253 + 0.438i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.827 - 0.561i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.827 - 0.561i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.88174 + 0.578164i\)
\(L(\frac12)\) \(\approx\) \(1.88174 + 0.578164i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.328 + 2.21i)T \)
7 \( 1 + (0.629 - 2.56i)T \)
good3 \( 1 + (-2.63 - 1.52i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (1.60 - 2.77i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 6.25iT - 13T^{2} \)
17 \( 1 + (-1.54 - 0.890i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.83 + 3.18i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.78 + 1.60i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.88T + 29T^{2} \)
31 \( 1 + (1.40 - 2.44i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.40 - 2.54i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 0.899T + 41T^{2} \)
43 \( 1 + 7.26iT - 43T^{2} \)
47 \( 1 + (-2.18 + 1.26i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-7.82 - 4.52i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.79 - 4.83i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.45 + 9.44i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.21 - 0.700i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 15.3T + 71T^{2} \)
73 \( 1 + (-6.07 - 3.50i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.0933 - 0.161i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.134iT - 83T^{2} \)
89 \( 1 + (-9.21 - 15.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 9.75iT - 97T^{2} \)
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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

−12.32778739993848647362643342155, −10.57524801698074442212435418326, −9.873401697097683972644317883965, −8.949035932211813826165004719869, −8.469150098693691410309838782683, −7.51099135493906297885707619697, −5.51622149761302727639856614959, −4.73130633959552641205910178262, −3.32572921940888910591880597172, −2.26196316199171495167961399091, 1.79797385274551732339830240413, 3.09291390186334913077742172913, 3.95873433657663708949637017501, 6.22965102620369584524498194679, 7.13268843027534865890813690661, 7.69299084692066168907628482331, 8.842246273942315225916954364669, 9.709820135319829386845045850346, 10.76546106410069265354665709777, 11.81269030532021011007810334925

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