Properties

Label 2-285-15.2-c1-0-11
Degree $2$
Conductor $285$
Sign $-0.124 - 0.992i$
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  + (0.0307 − 0.0307i)2-s + (0.939 + 1.45i)3-s + 1.99i·4-s + (1.20 + 1.88i)5-s + (0.0737 + 0.0158i)6-s + (−0.715 − 0.715i)7-s + (0.123 + 0.123i)8-s + (−1.23 + 2.73i)9-s + (0.0950 + 0.0209i)10-s − 5.75i·11-s + (−2.90 + 1.87i)12-s + (−1.65 + 1.65i)13-s − 0.0440·14-s + (−1.61 + 3.52i)15-s − 3.98·16-s + (4.22 − 4.22i)17-s + ⋯
L(s)  = 1  + (0.0217 − 0.0217i)2-s + (0.542 + 0.840i)3-s + 0.999i·4-s + (0.538 + 0.842i)5-s + (0.0301 + 0.00647i)6-s + (−0.270 − 0.270i)7-s + (0.0435 + 0.0435i)8-s + (−0.411 + 0.911i)9-s + (0.0300 + 0.00663i)10-s − 1.73i·11-s + (−0.839 + 0.542i)12-s + (−0.457 + 0.457i)13-s − 0.0117·14-s + (−0.416 + 0.909i)15-s − 0.997·16-s + (1.02 − 1.02i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.124 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.124 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $-0.124 - 0.992i$
Analytic conductor: \(2.27573\)
Root analytic conductor: \(1.50855\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{285} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 285,\ (\ :1/2),\ -0.124 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03913 + 1.17713i\)
\(L(\frac12)\) \(\approx\) \(1.03913 + 1.17713i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.939 - 1.45i)T \)
5 \( 1 + (-1.20 - 1.88i)T \)
19 \( 1 + iT \)
good2 \( 1 + (-0.0307 + 0.0307i)T - 2iT^{2} \)
7 \( 1 + (0.715 + 0.715i)T + 7iT^{2} \)
11 \( 1 + 5.75iT - 11T^{2} \)
13 \( 1 + (1.65 - 1.65i)T - 13iT^{2} \)
17 \( 1 + (-4.22 + 4.22i)T - 17iT^{2} \)
23 \( 1 + (-1.96 - 1.96i)T + 23iT^{2} \)
29 \( 1 - 5.38T + 29T^{2} \)
31 \( 1 - 4.85T + 31T^{2} \)
37 \( 1 + (-6.33 - 6.33i)T + 37iT^{2} \)
41 \( 1 + 1.84iT - 41T^{2} \)
43 \( 1 + (1.35 - 1.35i)T - 43iT^{2} \)
47 \( 1 + (5.93 - 5.93i)T - 47iT^{2} \)
53 \( 1 + (-0.615 - 0.615i)T + 53iT^{2} \)
59 \( 1 - 2.01T + 59T^{2} \)
61 \( 1 + 5.22T + 61T^{2} \)
67 \( 1 + (8.61 + 8.61i)T + 67iT^{2} \)
71 \( 1 + 8.55iT - 71T^{2} \)
73 \( 1 + (8.09 - 8.09i)T - 73iT^{2} \)
79 \( 1 + 13.6iT - 79T^{2} \)
83 \( 1 + (-1.52 - 1.52i)T + 83iT^{2} \)
89 \( 1 + 7.80T + 89T^{2} \)
97 \( 1 + (-9.06 - 9.06i)T + 97iT^{2} \)
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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

−11.80266988731153778336803620924, −11.16315049007698138980532145871, −10.11181271947194490764954381775, −9.297017059937571963215283441144, −8.311042942215874817178986331428, −7.37835418443740436559807491645, −6.17003278992011606549435621601, −4.75092791787013353542711270964, −3.25524777189915372283649092144, −2.91160242453822348867507564398, 1.28408943661657294269167615456, 2.42287796971986774365051821152, 4.51485650748165135480281016940, 5.65166773518336377924695171697, 6.54739562452800765634639249240, 7.70460366259442757540189992670, 8.782235503848953711286439527426, 9.770046649217751754637516901828, 10.20469460471893703864886198269, 11.97435526101412070344757320700

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