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Properties

Label 2-285-5.4-c1-0-1
Degree $2$
Conductor $285$
Sign $-0.856 + 0.515i$
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  + 2.73i·2-s − i·3-s − 5.48·4-s + (1.15 + 1.91i)5-s + 2.73·6-s + 2.95i·7-s − 9.52i·8-s − 9-s + (−5.23 + 3.15i)10-s − 4.70·11-s + 5.48i·12-s + 3.69i·13-s − 8.08·14-s + (1.91 − 1.15i)15-s + 15.0·16-s − 4.86i·17-s + ⋯
L(s)  = 1  + 1.93i·2-s − 0.577i·3-s − 2.74·4-s + (0.515 + 0.856i)5-s + 1.11·6-s + 1.11i·7-s − 3.36i·8-s − 0.333·9-s + (−1.65 + 0.997i)10-s − 1.41·11-s + 1.58i·12-s + 1.02i·13-s − 2.15·14-s + (0.494 − 0.297i)15-s + 3.77·16-s − 1.18i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.856 + 0.515i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.856 + 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $-0.856 + 0.515i$
Analytic conductor: \(2.27573\)
Root analytic conductor: \(1.50855\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{285} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 285,\ (\ :1/2),\ -0.856 + 0.515i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.238327 - 0.857776i\)
\(L(\frac12)\) \(\approx\) \(0.238327 - 0.857776i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
5 \( 1 + (-1.15 - 1.91i)T \)
19 \( 1 + T \)
good2 \( 1 - 2.73iT - 2T^{2} \)
7 \( 1 - 2.95iT - 7T^{2} \)
11 \( 1 + 4.70T + 11T^{2} \)
13 \( 1 - 3.69iT - 13T^{2} \)
17 \( 1 + 4.86iT - 17T^{2} \)
23 \( 1 - 2.38iT - 23T^{2} \)
29 \( 1 - 6.56T + 29T^{2} \)
31 \( 1 - 1.78T + 31T^{2} \)
37 \( 1 - 3.73iT - 37T^{2} \)
41 \( 1 - 2.39T + 41T^{2} \)
43 \( 1 - 8.82iT - 43T^{2} \)
47 \( 1 - 1.48iT - 47T^{2} \)
53 \( 1 - 1.31iT - 53T^{2} \)
59 \( 1 - 2.04T + 59T^{2} \)
61 \( 1 - 13.2T + 61T^{2} \)
67 \( 1 - 5.51iT - 67T^{2} \)
71 \( 1 + 5.51T + 71T^{2} \)
73 \( 1 + 5.57iT - 73T^{2} \)
79 \( 1 - 5.18T + 79T^{2} \)
83 \( 1 + 6.83iT - 83T^{2} \)
89 \( 1 + 7.13T + 89T^{2} \)
97 \( 1 - 9.88iT - 97T^{2} \)
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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

−12.83641317182232678956289006356, −11.60957670350199703578053140493, −10.04551828813565894050657036205, −9.185145530808219640206491299681, −8.244035832311476932576581079685, −7.33702513454565913423137626084, −6.53551892405435960096646131896, −5.72806247375908558371554042868, −4.84478564568619618892603941997, −2.74422248891734888780443794037, 0.68407012156501779789114768567, 2.40037960844451520328229655687, 3.74426391937073545867058546785, 4.69389794394798109417910174849, 5.56762009884054602221292145967, 8.065320432539846242668162391853, 8.688543217095080511682689181431, 10.07355109887880194382114180152, 10.28023643610802142666712611965, 10.93387428588830651605735449877

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