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Properties

Label 2-285-19.11-c1-0-0
Degree $2$
Conductor $285$
Sign $-0.999 - 0.0134i$
Analytic cond. $2.27573$
Root an. cond. $1.50855$
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  + (−0.690 + 1.19i)2-s + (0.5 − 0.866i)3-s + (0.0458 + 0.0794i)4-s + (−0.5 + 0.866i)5-s + (0.690 + 1.19i)6-s − 4.36·7-s − 2.88·8-s + (−0.499 − 0.866i)9-s + (−0.690 − 1.19i)10-s − 4.31·11-s + 0.0917·12-s + (3.18 + 5.51i)13-s + (3.01 − 5.21i)14-s + (0.499 + 0.866i)15-s + (1.90 − 3.29i)16-s + (−2.85 + 4.95i)17-s + ⋯
L(s)  = 1  + (−0.488 + 0.845i)2-s + (0.288 − 0.499i)3-s + (0.0229 + 0.0397i)4-s + (−0.223 + 0.387i)5-s + (0.281 + 0.488i)6-s − 1.64·7-s − 1.02·8-s + (−0.166 − 0.288i)9-s + (−0.218 − 0.378i)10-s − 1.30·11-s + 0.0264·12-s + (0.882 + 1.52i)13-s + (0.805 − 1.39i)14-s + (0.129 + 0.223i)15-s + (0.476 − 0.824i)16-s + (−0.693 + 1.20i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0134i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0134i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $-0.999 - 0.0134i$
Analytic conductor: \(2.27573\)
Root analytic conductor: \(1.50855\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{285} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 285,\ (\ :1/2),\ -0.999 - 0.0134i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00325016 + 0.481957i\)
\(L(\frac12)\) \(\approx\) \(0.00325016 + 0.481957i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-2.97 + 3.18i)T \)
good2 \( 1 + (0.690 - 1.19i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + 4.36T + 7T^{2} \)
11 \( 1 + 4.31T + 11T^{2} \)
13 \( 1 + (-3.18 - 5.51i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.85 - 4.95i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-0.289 - 0.501i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.77 - 3.07i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.18T + 31T^{2} \)
37 \( 1 + 6.54T + 37T^{2} \)
41 \( 1 + (-0.381 + 0.660i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.18 + 5.51i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.36 - 2.36i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.56 + 4.44i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.91 - 3.31i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.01 - 10.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.00 - 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.53 - 2.66i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.04 - 7.00i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.66 - 9.81i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.67T + 83T^{2} \)
89 \( 1 + (4.92 + 8.52i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.48204026971624564421792672036, −11.38376655319340634601159416434, −10.22323263023752149772139185112, −9.078902200386207614240142205940, −8.484247927469516849374656310011, −7.18317465114586838691270875020, −6.74568410035415597238812530188, −5.85915779471392451122374086296, −3.72636837396842573205641986963, −2.64598613641558642873812438999, 0.37912489363981368549273333467, 2.76525074677367621382537737091, 3.40581146713300206575479202280, 5.25641254999717448318936521705, 6.23551915807865749724830588306, 7.77219905976077388957667120438, 8.838994959862138465354530825694, 9.734300788430909066657903763165, 10.27694102695172549069035213511, 11.06643024845007609009211663391

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