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Properties

Label 2-270-135.122-c1-0-12
Degree $2$
Conductor $270$
Sign $0.490 - 0.871i$
Analytic cond. $2.15596$
Root an. cond. $1.46831$
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  + (0.819 + 0.573i)2-s + (1.17 + 1.27i)3-s + (0.342 + 0.939i)4-s + (2.18 − 0.473i)5-s + (0.234 + 1.71i)6-s + (−0.171 − 0.368i)7-s + (−0.258 + 0.965i)8-s + (−0.232 + 2.99i)9-s + (2.06 + 0.865i)10-s + (−3.16 − 3.76i)11-s + (−0.792 + 1.54i)12-s + (−2.75 − 3.93i)13-s + (0.0706 − 0.400i)14-s + (3.17 + 2.22i)15-s + (−0.766 + 0.642i)16-s + (0.352 + 1.31i)17-s + ⋯
L(s)  = 1  + (0.579 + 0.405i)2-s + (0.679 + 0.733i)3-s + (0.171 + 0.469i)4-s + (0.977 − 0.211i)5-s + (0.0956 + 0.700i)6-s + (−0.0649 − 0.139i)7-s + (−0.0915 + 0.341i)8-s + (−0.0774 + 0.996i)9-s + (0.652 + 0.273i)10-s + (−0.952 − 1.13i)11-s + (−0.228 + 0.444i)12-s + (−0.764 − 1.09i)13-s + (0.0188 − 0.107i)14-s + (0.819 + 0.573i)15-s + (−0.191 + 0.160i)16-s + (0.0856 + 0.319i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.490 - 0.871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.490 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(270\)    =    \(2 \cdot 3^{3} \cdot 5\)
Sign: $0.490 - 0.871i$
Analytic conductor: \(2.15596\)
Root analytic conductor: \(1.46831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{270} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 270,\ (\ :1/2),\ 0.490 - 0.871i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.90736 + 1.11504i\)
\(L(\frac12)\) \(\approx\) \(1.90736 + 1.11504i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.819 - 0.573i)T \)
3 \( 1 + (-1.17 - 1.27i)T \)
5 \( 1 + (-2.18 + 0.473i)T \)
good7 \( 1 + (0.171 + 0.368i)T + (-4.49 + 5.36i)T^{2} \)
11 \( 1 + (3.16 + 3.76i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (2.75 + 3.93i)T + (-4.44 + 12.2i)T^{2} \)
17 \( 1 + (-0.352 - 1.31i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.763 - 0.440i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.893 - 0.416i)T + (14.7 + 17.6i)T^{2} \)
29 \( 1 + (-1.09 - 6.18i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (7.20 - 2.62i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (8.35 - 2.23i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-10.9 - 1.92i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (-1.07 + 12.3i)T + (-42.3 - 7.46i)T^{2} \)
47 \( 1 + (-3.46 + 1.61i)T + (30.2 - 36.0i)T^{2} \)
53 \( 1 + (-0.890 - 0.890i)T + 53iT^{2} \)
59 \( 1 + (-0.0286 - 0.0240i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-3.45 - 1.25i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (2.00 - 1.40i)T + (22.9 - 62.9i)T^{2} \)
71 \( 1 + (-4.92 - 2.84i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.355 + 0.0951i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (1.82 - 0.320i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (0.186 - 0.266i)T + (-28.3 - 77.9i)T^{2} \)
89 \( 1 + (-6.26 - 10.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.287 + 0.0251i)T + (95.5 + 16.8i)T^{2} \)
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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

−12.48726761372495364008288836324, −10.71649317437619983903581156554, −10.38590287120999569433132507395, −9.070992679465225110164108122429, −8.317323770137752944999344410746, −7.20837496316792061314031825765, −5.56505765236554046644749547674, −5.20938901761131584948499824485, −3.56303952127864022751643687507, −2.51544014532453319085730895662, 1.97479199676417637185191795091, 2.65818957569091392614577071198, 4.41342513147953641087639261867, 5.69215031133282030180467067678, 6.82860966668357853227825787768, 7.61320794239827364030966199276, 9.241899655048744947287056178852, 9.694601151466485484074959436851, 10.87596754724279582910511605993, 12.12772095844040753247096053440

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