Properties

Label 2-285-15.2-c1-0-28
Degree $2$
Conductor $285$
Sign $-0.800 + 0.599i$
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  + (−1.91 + 1.91i)2-s + (−0.520 − 1.65i)3-s − 5.34i·4-s + (0.478 − 2.18i)5-s + (4.16 + 2.16i)6-s + (−1.54 − 1.54i)7-s + (6.41 + 6.41i)8-s + (−2.45 + 1.72i)9-s + (3.26 + 5.10i)10-s − 2.77i·11-s + (−8.82 + 2.78i)12-s + (−3.18 + 3.18i)13-s + 5.93·14-s + (−3.85 + 0.347i)15-s − 13.8·16-s + (−3.06 + 3.06i)17-s + ⋯
L(s)  = 1  + (−1.35 + 1.35i)2-s + (−0.300 − 0.953i)3-s − 2.67i·4-s + (0.213 − 0.976i)5-s + (1.69 + 0.885i)6-s + (−0.585 − 0.585i)7-s + (2.26 + 2.26i)8-s + (−0.819 + 0.573i)9-s + (1.03 + 1.61i)10-s − 0.836i·11-s + (−2.54 + 0.803i)12-s + (−0.884 + 0.884i)13-s + 1.58·14-s + (−0.995 + 0.0897i)15-s − 3.46·16-s + (−0.744 + 0.744i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.800 + 0.599i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.800 + 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $-0.800 + 0.599i$
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: \((2,\ 285,\ (\ :1/2),\ -0.800 + 0.599i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0627419 - 0.188242i\)
\(L(\frac12)\) \(\approx\) \(0.0627419 - 0.188242i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.520 + 1.65i)T \)
5 \( 1 + (-0.478 + 2.18i)T \)
19 \( 1 - iT \)
good2 \( 1 + (1.91 - 1.91i)T - 2iT^{2} \)
7 \( 1 + (1.54 + 1.54i)T + 7iT^{2} \)
11 \( 1 + 2.77iT - 11T^{2} \)
13 \( 1 + (3.18 - 3.18i)T - 13iT^{2} \)
17 \( 1 + (3.06 - 3.06i)T - 17iT^{2} \)
23 \( 1 + (-1.05 - 1.05i)T + 23iT^{2} \)
29 \( 1 - 0.341T + 29T^{2} \)
31 \( 1 - 6.05T + 31T^{2} \)
37 \( 1 + (5.33 + 5.33i)T + 37iT^{2} \)
41 \( 1 + 0.460iT - 41T^{2} \)
43 \( 1 + (3.43 - 3.43i)T - 43iT^{2} \)
47 \( 1 + (-1.93 + 1.93i)T - 47iT^{2} \)
53 \( 1 + (4.19 + 4.19i)T + 53iT^{2} \)
59 \( 1 + 12.0T + 59T^{2} \)
61 \( 1 + 12.4T + 61T^{2} \)
67 \( 1 + (2.96 + 2.96i)T + 67iT^{2} \)
71 \( 1 - 9.44iT - 71T^{2} \)
73 \( 1 + (-11.0 + 11.0i)T - 73iT^{2} \)
79 \( 1 - 1.95iT - 79T^{2} \)
83 \( 1 + (1.98 + 1.98i)T + 83iT^{2} \)
89 \( 1 - 2.63T + 89T^{2} \)
97 \( 1 + (7.00 + 7.00i)T + 97iT^{2} \)
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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

−11.21094185248778653334123755218, −10.17572229972920905597967044498, −9.171245019500982337350717888244, −8.472068241314308838159923180009, −7.58974149765807765209154515940, −6.64072542753284542100562927281, −5.98621100945657427225498080635, −4.82010999960552267092009311247, −1.68080470178526485380632049063, −0.23187970526397696969078672502, 2.53232474817516235309881234443, 3.16766343465129340006987986090, 4.71281839504139866270363002085, 6.55646630478705924973955584919, 7.68157576309244000921879509123, 8.983381647884655683091878494624, 9.664187518067004176276231402137, 10.25520694628231502692472103694, 10.92300821188777658186897160627, 11.89680597336349802854607423577

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