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Properties

Label 2-285-15.2-c1-0-12
Degree $2$
Conductor $285$
Sign $0.999 + 0.0299i$
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 − i)2-s − 1.73·3-s + (1.86 + 1.23i)5-s + (−1.73 + 1.73i)6-s + (−0.633 − 0.633i)7-s + (2 + 2i)8-s + 2.99·9-s + (3.09 − 0.633i)10-s + 3.73i·11-s + (−0.267 + 0.267i)13-s − 1.26·14-s + (−3.23 − 2.13i)15-s + 4·16-s + (5.09 − 5.09i)17-s + (2.99 − 2.99i)18-s − i·19-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s − 1.00·3-s + (0.834 + 0.550i)5-s + (−0.707 + 0.707i)6-s + (−0.239 − 0.239i)7-s + (0.707 + 0.707i)8-s + 0.999·9-s + (0.979 − 0.200i)10-s + 1.12i·11-s + (−0.0743 + 0.0743i)13-s − 0.338·14-s + (−0.834 − 0.550i)15-s + 16-s + (1.23 − 1.23i)17-s + (0.707 − 0.707i)18-s − 0.229i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0299i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0299i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $0.999 + 0.0299i$
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: \((2,\ 285,\ (\ :1/2),\ 0.999 + 0.0299i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58443 - 0.0237489i\)
\(L(\frac12)\) \(\approx\) \(1.58443 - 0.0237489i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.73T \)
5 \( 1 + (-1.86 - 1.23i)T \)
19 \( 1 + iT \)
good2 \( 1 + (-1 + i)T - 2iT^{2} \)
7 \( 1 + (0.633 + 0.633i)T + 7iT^{2} \)
11 \( 1 - 3.73iT - 11T^{2} \)
13 \( 1 + (0.267 - 0.267i)T - 13iT^{2} \)
17 \( 1 + (-5.09 + 5.09i)T - 17iT^{2} \)
23 \( 1 + (0.464 + 0.464i)T + 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 5.26T + 31T^{2} \)
37 \( 1 + (4.46 + 4.46i)T + 37iT^{2} \)
41 \( 1 - 9.66iT - 41T^{2} \)
43 \( 1 + (5.09 - 5.09i)T - 43iT^{2} \)
47 \( 1 + (-8.56 + 8.56i)T - 47iT^{2} \)
53 \( 1 + (5.73 + 5.73i)T + 53iT^{2} \)
59 \( 1 - 0.196T + 59T^{2} \)
61 \( 1 + 7.73T + 61T^{2} \)
67 \( 1 + (2.19 + 2.19i)T + 67iT^{2} \)
71 \( 1 + 10.7iT - 71T^{2} \)
73 \( 1 + (-5.36 + 5.36i)T - 73iT^{2} \)
79 \( 1 - 10.3iT - 79T^{2} \)
83 \( 1 + (-0.267 - 0.267i)T + 83iT^{2} \)
89 \( 1 - 7.26T + 89T^{2} \)
97 \( 1 + (6.46 + 6.46i)T + 97iT^{2} \)
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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

−11.91330604906989153183146094419, −11.05108964837280563735504728694, −10.17735656638959860906632664798, −9.533455987053645865293904629008, −7.55737316725042753269333811778, −6.83323683579304600148276672637, −5.49589591726754967867508046147, −4.71883337381461855929368964241, −3.31641098604596688722250899523, −1.86888416959089834765377201080, 1.32002361109265517263088519616, 3.81757130937222182792840855495, 5.19191569403087159417245125499, 5.79783389064441142697682829037, 6.33273612252533784419462638470, 7.66134005947644792797317383994, 9.041941247054608985735511418182, 10.19385245643673848885561472458, 10.72264878054662639647389177703, 12.19484247834924103682269733764

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