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Properties

Label 2-280-35.13-c1-0-5
Degree $2$
Conductor $280$
Sign $0.700 + 0.713i$
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  + (−2.16 + 2.16i)3-s + (0.272 − 2.21i)5-s + (−1.63 − 2.07i)7-s − 6.41i·9-s + 2.56·11-s + (1.35 − 1.35i)13-s + (4.22 + 5.40i)15-s + (2.10 + 2.10i)17-s + 4.43·19-s + (8.06 + 0.962i)21-s + (−5.78 − 5.78i)23-s + (−4.85 − 1.21i)25-s + (7.41 + 7.41i)27-s − 4.28i·29-s − 9.70i·31-s + ⋯
L(s)  = 1  + (−1.25 + 1.25i)3-s + (0.121 − 0.992i)5-s + (−0.618 − 0.785i)7-s − 2.13i·9-s + 0.774·11-s + (0.375 − 0.375i)13-s + (1.09 + 1.39i)15-s + (0.509 + 0.509i)17-s + 1.01·19-s + (1.75 + 0.210i)21-s + (−1.20 − 1.20i)23-s + (−0.970 − 0.242i)25-s + (1.42 + 1.42i)27-s − 0.795i·29-s − 1.74i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.700 + 0.713i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.700 + 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.700 + 0.713i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.700 + 0.713i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.678496 - 0.284732i\)
\(L(\frac12)\) \(\approx\) \(0.678496 - 0.284732i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.272 + 2.21i)T \)
7 \( 1 + (1.63 + 2.07i)T \)
good3 \( 1 + (2.16 - 2.16i)T - 3iT^{2} \)
11 \( 1 - 2.56T + 11T^{2} \)
13 \( 1 + (-1.35 + 1.35i)T - 13iT^{2} \)
17 \( 1 + (-2.10 - 2.10i)T + 17iT^{2} \)
19 \( 1 - 4.43T + 19T^{2} \)
23 \( 1 + (5.78 + 5.78i)T + 23iT^{2} \)
29 \( 1 + 4.28iT - 29T^{2} \)
31 \( 1 + 9.70iT - 31T^{2} \)
37 \( 1 + (-0.183 + 0.183i)T - 37iT^{2} \)
41 \( 1 + 3.81iT - 41T^{2} \)
43 \( 1 + (-5.86 - 5.86i)T + 43iT^{2} \)
47 \( 1 + (1.83 + 1.83i)T + 47iT^{2} \)
53 \( 1 + (2.38 + 2.38i)T + 53iT^{2} \)
59 \( 1 + 6.34T + 59T^{2} \)
61 \( 1 - 8.62iT - 61T^{2} \)
67 \( 1 + (-6.68 + 6.68i)T - 67iT^{2} \)
71 \( 1 + 6.04T + 71T^{2} \)
73 \( 1 + (-1.88 + 1.88i)T - 73iT^{2} \)
79 \( 1 - 12.2iT - 79T^{2} \)
83 \( 1 + (2.08 - 2.08i)T - 83iT^{2} \)
89 \( 1 - 10.1T + 89T^{2} \)
97 \( 1 + (-7.15 - 7.15i)T + 97iT^{2} \)
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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

−11.71738842409640193062443796697, −10.70524599113440813125094186342, −9.850349652913695761548758064248, −9.370403120433867923480522917276, −7.953359004023207456326492807314, −6.30546926599520154168283829717, −5.69792246543344137468938559407, −4.41025917620075334831530846150, −3.81442270040069701138461445511, −0.69451969013946289930915046370, 1.63806601430950233559082237624, 3.25972481538477026051717277976, 5.37074777300273136442872925252, 6.15351496899180379526792935069, 6.86722444900597480047107730974, 7.67434990034137953489791836892, 9.224554368505187862183876650391, 10.30601332612305097473707234344, 11.43679296078032137066814684215, 11.86177047665930360369304492599

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