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Properties

Label 2-285-285.23-c1-0-27
Degree $2$
Conductor $285$
Sign $-0.922 + 0.384i$
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  + (−0.425 − 0.911i)2-s + (−1.70 + 0.283i)3-s + (0.635 − 0.757i)4-s + (0.685 − 2.12i)5-s + (0.984 + 1.43i)6-s + (−0.0170 − 0.0637i)7-s + (−2.90 − 0.777i)8-s + (2.83 − 0.968i)9-s + (−2.23 + 0.279i)10-s + (−0.508 + 0.293i)11-s + (−0.871 + 1.47i)12-s + (−0.808 + 0.566i)13-s + (−0.0508 + 0.0426i)14-s + (−0.568 + 3.83i)15-s + (0.181 + 1.02i)16-s + (−1.25 − 2.69i)17-s + ⋯
L(s)  = 1  + (−0.300 − 0.644i)2-s + (−0.986 + 0.163i)3-s + (0.317 − 0.378i)4-s + (0.306 − 0.951i)5-s + (0.401 + 0.586i)6-s + (−0.00645 − 0.0240i)7-s + (−1.02 − 0.275i)8-s + (0.946 − 0.322i)9-s + (−0.705 + 0.0885i)10-s + (−0.153 + 0.0884i)11-s + (−0.251 + 0.425i)12-s + (−0.224 + 0.157i)13-s + (−0.0135 + 0.0113i)14-s + (−0.146 + 0.989i)15-s + (0.0453 + 0.257i)16-s + (−0.305 − 0.654i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.922 + 0.384i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.922 + 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $-0.922 + 0.384i$
Analytic conductor: \(2.27573\)
Root analytic conductor: \(1.50855\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{285} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 285,\ (\ :1/2),\ -0.922 + 0.384i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.141319 - 0.705957i\)
\(L(\frac12)\) \(\approx\) \(0.141319 - 0.705957i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.70 - 0.283i)T \)
5 \( 1 + (-0.685 + 2.12i)T \)
19 \( 1 + (1.84 + 3.94i)T \)
good2 \( 1 + (0.425 + 0.911i)T + (-1.28 + 1.53i)T^{2} \)
7 \( 1 + (0.0170 + 0.0637i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (0.508 - 0.293i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.808 - 0.566i)T + (4.44 - 12.2i)T^{2} \)
17 \( 1 + (1.25 + 2.69i)T + (-10.9 + 13.0i)T^{2} \)
23 \( 1 + (3.65 - 0.319i)T + (22.6 - 3.99i)T^{2} \)
29 \( 1 + (2.64 - 0.962i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (1.34 - 2.32i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.97 + 4.97i)T - 37iT^{2} \)
41 \( 1 + (-9.04 + 1.59i)T + (38.5 - 14.0i)T^{2} \)
43 \( 1 + (-8.99 - 0.786i)T + (42.3 + 7.46i)T^{2} \)
47 \( 1 + (-2.88 - 1.34i)T + (30.2 + 36.0i)T^{2} \)
53 \( 1 + (-0.904 + 0.0791i)T + (52.1 - 9.20i)T^{2} \)
59 \( 1 + (7.64 + 2.78i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (0.829 + 0.695i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (0.965 - 2.06i)T + (-43.0 - 51.3i)T^{2} \)
71 \( 1 + (-4.86 - 5.79i)T + (-12.3 + 69.9i)T^{2} \)
73 \( 1 + (-8.16 + 11.6i)T + (-24.9 - 68.5i)T^{2} \)
79 \( 1 + (-6.16 + 1.08i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (0.451 + 1.68i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (0.572 - 3.24i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (13.4 - 6.28i)T + (62.3 - 74.3i)T^{2} \)
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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

−11.28355478870249816993998004710, −10.66268564216989230988197705447, −9.573706195392039508332028406123, −9.095076704924693797797356585070, −7.40046979560130375055802836339, −6.21217385178901137539296136803, −5.37136508492005165112746795693, −4.26990467721324761725882981323, −2.19991558506295574284667315492, −0.64694030105976230763358946743, 2.32581006209058695067727328546, 3.98908230534403937811702126373, 5.84952822440838667447384564206, 6.20846866621541956215813404369, 7.33010409925614405584307910478, 7.985588075042003520269142332270, 9.484923595275480788256882013392, 10.53913769141198353756926016525, 11.20415747183228157942435031751, 12.17313856209088789545563432409

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