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Properties

Label 2-285-285.68-c1-0-9
Degree $2$
Conductor $285$
Sign $-0.257 - 0.966i$
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  + (−1.51 + 0.405i)2-s + (−1.64 + 0.556i)3-s + (0.389 − 0.224i)4-s + (0.718 + 2.11i)5-s + (2.25 − 1.50i)6-s + (2.61 − 2.61i)7-s + (1.71 − 1.71i)8-s + (2.37 − 1.82i)9-s + (−1.94 − 2.90i)10-s + 5.27i·11-s + (−0.513 + 0.585i)12-s + (0.467 − 1.74i)13-s + (−2.89 + 5.01i)14-s + (−2.35 − 3.07i)15-s + (−2.34 + 4.06i)16-s + (−2.61 + 0.701i)17-s + ⋯
L(s)  = 1  + (−1.06 + 0.286i)2-s + (−0.946 + 0.321i)3-s + (0.194 − 0.112i)4-s + (0.321 + 0.946i)5-s + (0.920 − 0.614i)6-s + (0.988 − 0.988i)7-s + (0.606 − 0.606i)8-s + (0.793 − 0.608i)9-s + (−0.614 − 0.920i)10-s + 1.59i·11-s + (−0.148 + 0.168i)12-s + (0.129 − 0.484i)13-s + (−0.773 + 1.34i)14-s + (−0.608 − 0.793i)15-s + (−0.587 + 1.01i)16-s + (−0.635 + 0.170i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.257 - 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.257 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $-0.257 - 0.966i$
Analytic conductor: \(2.27573\)
Root analytic conductor: \(1.50855\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{285} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 285,\ (\ :1/2),\ -0.257 - 0.966i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.340699 + 0.443430i\)
\(L(\frac12)\) \(\approx\) \(0.340699 + 0.443430i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.64 - 0.556i)T \)
5 \( 1 + (-0.718 - 2.11i)T \)
19 \( 1 + (0.232 - 4.35i)T \)
good2 \( 1 + (1.51 - 0.405i)T + (1.73 - i)T^{2} \)
7 \( 1 + (-2.61 + 2.61i)T - 7iT^{2} \)
11 \( 1 - 5.27iT - 11T^{2} \)
13 \( 1 + (-0.467 + 1.74i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (2.61 - 0.701i)T + (14.7 - 8.5i)T^{2} \)
23 \( 1 + (-4.20 - 1.12i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (0.173 + 0.300i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.90T + 31T^{2} \)
37 \( 1 + (5.09 - 5.09i)T - 37iT^{2} \)
41 \( 1 + (8.19 + 4.73i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.11 + 0.299i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (2.00 - 7.47i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (1.41 + 0.378i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-0.522 + 0.905i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.24 - 10.8i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.09 + 0.561i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-9.10 - 5.25i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-8.34 + 2.23i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-8.23 - 4.75i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.49 + 2.49i)T - 83iT^{2} \)
89 \( 1 + (4.59 + 7.96i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.469 - 1.75i)T + (-84.0 + 48.5i)T^{2} \)
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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

−11.76048436922586502825639971497, −10.67243963250139845696793657604, −10.31899959135189788388434004653, −9.592119959914287532354519486958, −8.109934497243333804855699678723, −7.20904399907670353846328872403, −6.61829835271406062618488264584, −4.97548747714265306089676861551, −3.99901206000409802630606379986, −1.50405028742012120305221895757, 0.75845360188528259856732243088, 2.06077439856435696863989853250, 4.81438092003204519625220523138, 5.33531836925037652970112816985, 6.59957168271502075298419543510, 8.231438497746438180782146638856, 8.627499684695440969408593519035, 9.502972436269717260532333525062, 10.84656059069451312313816043692, 11.34648046698193213539134062165

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