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Properties

Label 2-285-15.2-c1-0-14
Degree $2$
Conductor $285$
Sign $0.890 - 0.455i$
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  + (−1.82 + 1.82i)2-s + (−1.28 + 1.16i)3-s − 4.64i·4-s + (1.53 + 1.63i)5-s + (0.217 − 4.45i)6-s + (−1.83 − 1.83i)7-s + (4.81 + 4.81i)8-s + (0.291 − 2.98i)9-s + (−5.75 − 0.181i)10-s − 4.54i·11-s + (5.40 + 5.95i)12-s + (4.46 − 4.46i)13-s + 6.69·14-s + (−3.86 − 0.310i)15-s − 8.25·16-s + (−1.82 + 1.82i)17-s + ⋯
L(s)  = 1  + (−1.28 + 1.28i)2-s + (−0.740 + 0.671i)3-s − 2.32i·4-s + (0.684 + 0.729i)5-s + (0.0887 − 1.82i)6-s + (−0.694 − 0.694i)7-s + (1.70 + 1.70i)8-s + (0.0972 − 0.995i)9-s + (−1.82 − 0.0575i)10-s − 1.36i·11-s + (1.55 + 1.71i)12-s + (1.23 − 1.23i)13-s + 1.78·14-s + (−0.996 − 0.0802i)15-s − 2.06·16-s + (−0.443 + 0.443i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.455i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.890 - 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $0.890 - 0.455i$
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: \((2,\ 285,\ (\ :1/2),\ 0.890 - 0.455i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.451416 + 0.108863i\)
\(L(\frac12)\) \(\approx\) \(0.451416 + 0.108863i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.28 - 1.16i)T \)
5 \( 1 + (-1.53 - 1.63i)T \)
19 \( 1 + iT \)
good2 \( 1 + (1.82 - 1.82i)T - 2iT^{2} \)
7 \( 1 + (1.83 + 1.83i)T + 7iT^{2} \)
11 \( 1 + 4.54iT - 11T^{2} \)
13 \( 1 + (-4.46 + 4.46i)T - 13iT^{2} \)
17 \( 1 + (1.82 - 1.82i)T - 17iT^{2} \)
23 \( 1 + (3.81 + 3.81i)T + 23iT^{2} \)
29 \( 1 - 2.20T + 29T^{2} \)
31 \( 1 + 2.07T + 31T^{2} \)
37 \( 1 + (-0.958 - 0.958i)T + 37iT^{2} \)
41 \( 1 - 1.66iT - 41T^{2} \)
43 \( 1 + (-5.41 + 5.41i)T - 43iT^{2} \)
47 \( 1 + (-1.25 + 1.25i)T - 47iT^{2} \)
53 \( 1 + (7.89 + 7.89i)T + 53iT^{2} \)
59 \( 1 - 8.55T + 59T^{2} \)
61 \( 1 - 4.48T + 61T^{2} \)
67 \( 1 + (2.12 + 2.12i)T + 67iT^{2} \)
71 \( 1 + 3.02iT - 71T^{2} \)
73 \( 1 + (-3.69 + 3.69i)T - 73iT^{2} \)
79 \( 1 - 0.976iT - 79T^{2} \)
83 \( 1 + (0.264 + 0.264i)T + 83iT^{2} \)
89 \( 1 - 6.33T + 89T^{2} \)
97 \( 1 + (6.87 + 6.87i)T + 97iT^{2} \)
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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

−11.12887706692734125527514326981, −10.56187302820135927519614974418, −10.08720766669947618229769796184, −9.019145993524399711964224013787, −8.142226452708594505793846972599, −6.72320377313101777833459797393, −6.19279822373855497524953895707, −5.52910431470935544038422509525, −3.54328339754317904229389063754, −0.62753729514046662946302204162, 1.45035862395844092410057598079, 2.32129872748440316224311253800, 4.31048188232454499220673656976, 5.93755161248097201241019923078, 7.04036026241475897033513855663, 8.273042722996820694979864012479, 9.285733971712683078723212534778, 9.711120278783631776885920171479, 10.85731516290455804234249802396, 11.76336700630995751870243865877

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