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Properties

Label 2-285-15.2-c1-0-17
Degree $2$
Conductor $285$
Sign $0.948 - 0.315i$
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  + (0.913 − 0.913i)2-s + (1.44 + 0.951i)3-s + 0.331i·4-s + (−2.10 + 0.765i)5-s + (2.19 − 0.452i)6-s + (0.194 + 0.194i)7-s + (2.12 + 2.12i)8-s + (1.18 + 2.75i)9-s + (−1.21 + 2.61i)10-s − 0.417i·11-s + (−0.315 + 0.480i)12-s + (4.34 − 4.34i)13-s + 0.356·14-s + (−3.76 − 0.891i)15-s + 3.22·16-s + (−1.75 + 1.75i)17-s + ⋯
L(s)  = 1  + (0.645 − 0.645i)2-s + (0.835 + 0.549i)3-s + 0.165i·4-s + (−0.939 + 0.342i)5-s + (0.894 − 0.184i)6-s + (0.0736 + 0.0736i)7-s + (0.752 + 0.752i)8-s + (0.395 + 0.918i)9-s + (−0.385 + 0.827i)10-s − 0.125i·11-s + (−0.0912 + 0.138i)12-s + (1.20 − 1.20i)13-s + 0.0951·14-s + (−0.973 − 0.230i)15-s + 0.806·16-s + (−0.426 + 0.426i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.315i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 - 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $0.948 - 0.315i$
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: \((2,\ 285,\ (\ :1/2),\ 0.948 - 0.315i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.04503 + 0.331274i\)
\(L(\frac12)\) \(\approx\) \(2.04503 + 0.331274i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.44 - 0.951i)T \)
5 \( 1 + (2.10 - 0.765i)T \)
19 \( 1 + iT \)
good2 \( 1 + (-0.913 + 0.913i)T - 2iT^{2} \)
7 \( 1 + (-0.194 - 0.194i)T + 7iT^{2} \)
11 \( 1 + 0.417iT - 11T^{2} \)
13 \( 1 + (-4.34 + 4.34i)T - 13iT^{2} \)
17 \( 1 + (1.75 - 1.75i)T - 17iT^{2} \)
23 \( 1 + (3.99 + 3.99i)T + 23iT^{2} \)
29 \( 1 + 7.49T + 29T^{2} \)
31 \( 1 + 1.04T + 31T^{2} \)
37 \( 1 + (3.18 + 3.18i)T + 37iT^{2} \)
41 \( 1 + 4.47iT - 41T^{2} \)
43 \( 1 + (-1.64 + 1.64i)T - 43iT^{2} \)
47 \( 1 + (-6.22 + 6.22i)T - 47iT^{2} \)
53 \( 1 + (-7.09 - 7.09i)T + 53iT^{2} \)
59 \( 1 - 4.72T + 59T^{2} \)
61 \( 1 - 0.494T + 61T^{2} \)
67 \( 1 + (0.00945 + 0.00945i)T + 67iT^{2} \)
71 \( 1 + 6.32iT - 71T^{2} \)
73 \( 1 + (10.3 - 10.3i)T - 73iT^{2} \)
79 \( 1 - 3.38iT - 79T^{2} \)
83 \( 1 + (-4.89 - 4.89i)T + 83iT^{2} \)
89 \( 1 + 2.98T + 89T^{2} \)
97 \( 1 + (-6.77 - 6.77i)T + 97iT^{2} \)
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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

−11.89210909334777224559591722704, −10.81939035696904535976555541219, −10.51524322293657988887140240549, −8.764562853324643990759114958171, −8.211113603263646841903950894556, −7.28472061215707270802653403038, −5.47799609765450510644170349428, −4.04014592139567184890517382457, −3.61914162616713945022358670862, −2.39672467768159262718796888441, 1.53229202027818485640198097510, 3.67182250151275873677913109903, 4.39132128289729036347103072779, 5.91549047655113746748345490053, 6.95989845764248435813315317728, 7.67851484525433313462786798577, 8.744774634029551761821353633630, 9.604284008929589133325993431110, 11.08421722156080300112873219170, 11.94540938817583853945408368204

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