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Properties

Label 2-270-135.122-c1-0-3
Degree $2$
Conductor $270$
Sign $0.215 - 0.976i$
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 + (0.243 + 1.71i)3-s + (0.342 + 0.939i)4-s + (1.53 + 1.62i)5-s + (0.784 − 1.54i)6-s + (−0.164 − 0.351i)7-s + (0.258 − 0.965i)8-s + (−2.88 + 0.834i)9-s + (−0.323 − 2.21i)10-s + (3.24 + 3.86i)11-s + (−1.52 + 0.815i)12-s + (−2.12 − 3.03i)13-s + (−0.0674 + 0.382i)14-s + (−2.41 + 3.02i)15-s + (−0.766 + 0.642i)16-s + (−0.0721 − 0.269i)17-s + ⋯
L(s)  = 1  + (−0.579 − 0.405i)2-s + (0.140 + 0.990i)3-s + (0.171 + 0.469i)4-s + (0.685 + 0.727i)5-s + (0.320 − 0.630i)6-s + (−0.0620 − 0.133i)7-s + (0.0915 − 0.341i)8-s + (−0.960 + 0.278i)9-s + (−0.102 − 0.699i)10-s + (0.977 + 1.16i)11-s + (−0.441 + 0.235i)12-s + (−0.589 − 0.841i)13-s + (−0.0180 + 0.102i)14-s + (−0.624 + 0.781i)15-s + (−0.191 + 0.160i)16-s + (−0.0175 − 0.0653i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.215 - 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.215 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(270\)    =    \(2 \cdot 3^{3} \cdot 5\)
Sign: $0.215 - 0.976i$
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: \((2,\ 270,\ (\ :1/2),\ 0.215 - 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.823167 + 0.661479i\)
\(L(\frac12)\) \(\approx\) \(0.823167 + 0.661479i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.243 - 1.71i)T \)
5 \( 1 + (-1.53 - 1.62i)T \)
good7 \( 1 + (0.164 + 0.351i)T + (-4.49 + 5.36i)T^{2} \)
11 \( 1 + (-3.24 - 3.86i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (2.12 + 3.03i)T + (-4.44 + 12.2i)T^{2} \)
17 \( 1 + (0.0721 + 0.269i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (5.80 - 3.35i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.00 - 1.86i)T + (14.7 + 17.6i)T^{2} \)
29 \( 1 + (-0.711 - 4.03i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-8.64 + 3.14i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (-1.02 + 0.275i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-1.25 - 0.221i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (-0.614 + 7.02i)T + (-42.3 - 7.46i)T^{2} \)
47 \( 1 + (-0.552 + 0.257i)T + (30.2 - 36.0i)T^{2} \)
53 \( 1 + (7.96 + 7.96i)T + 53iT^{2} \)
59 \( 1 + (-3.03 - 2.54i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (9.72 + 3.54i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (3.80 - 2.66i)T + (22.9 - 62.9i)T^{2} \)
71 \( 1 + (-9.02 - 5.20i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.01 + 0.539i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-14.3 + 2.53i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (-4.36 + 6.23i)T + (-28.3 - 77.9i)T^{2} \)
89 \( 1 + (5.24 + 9.08i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.74 - 0.677i)T + (95.5 + 16.8i)T^{2} \)
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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

−11.89220545536746411207972611002, −10.81169401111214545367342171568, −10.13417390550719982385926780839, −9.603112601682339601118538444992, −8.593715276578659494876373858263, −7.31035091899050541862408058418, −6.20049951391967717379716394376, −4.74925815741028499656874775442, −3.47301490523921606049252157512, −2.19831451502929503964457465604, 1.03158093972102229185775664352, 2.50000962154309721454884121730, 4.68725472304984757115171385612, 6.24073973245735979070015668212, 6.50401203211132336075228316021, 7.980660940361684828603348105715, 8.887099249264118014424335402118, 9.296455776320119389647385957197, 10.80780642430372314198433453768, 11.79353169617142731482080681767

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