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Properties

Label 2-270-135.122-c1-0-7
Degree $2$
Conductor $270$
Sign $0.654 - 0.756i$
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 + (−1.72 + 0.169i)3-s + (0.342 + 0.939i)4-s + (2.07 − 0.845i)5-s + (−1.50 − 0.849i)6-s + (0.101 + 0.217i)7-s + (−0.258 + 0.965i)8-s + (2.94 − 0.585i)9-s + (2.18 + 0.494i)10-s + (2.07 + 2.47i)11-s + (−0.749 − 1.56i)12-s + (1.10 + 1.57i)13-s + (−0.0417 + 0.236i)14-s + (−3.42 + 1.80i)15-s + (−0.766 + 0.642i)16-s + (0.935 + 3.48i)17-s + ⋯
L(s)  = 1  + (0.579 + 0.405i)2-s + (−0.995 + 0.0980i)3-s + (0.171 + 0.469i)4-s + (0.925 − 0.378i)5-s + (−0.616 − 0.346i)6-s + (0.0383 + 0.0823i)7-s + (−0.0915 + 0.341i)8-s + (0.980 − 0.195i)9-s + (0.689 + 0.156i)10-s + (0.625 + 0.745i)11-s + (−0.216 − 0.450i)12-s + (0.306 + 0.437i)13-s + (−0.0111 + 0.0632i)14-s + (−0.884 + 0.466i)15-s + (−0.191 + 0.160i)16-s + (0.226 + 0.846i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 - 0.756i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.654 - 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(270\)    =    \(2 \cdot 3^{3} \cdot 5\)
Sign: $0.654 - 0.756i$
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: \((2,\ 270,\ (\ :1/2),\ 0.654 - 0.756i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39133 + 0.636105i\)
\(L(\frac12)\) \(\approx\) \(1.39133 + 0.636105i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.72 - 0.169i)T \)
5 \( 1 + (-2.07 + 0.845i)T \)
good7 \( 1 + (-0.101 - 0.217i)T + (-4.49 + 5.36i)T^{2} \)
11 \( 1 + (-2.07 - 2.47i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (-1.10 - 1.57i)T + (-4.44 + 12.2i)T^{2} \)
17 \( 1 + (-0.935 - 3.48i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.18 + 0.683i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.16 + 2.87i)T + (14.7 + 17.6i)T^{2} \)
29 \( 1 + (0.340 + 1.92i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-2.42 + 0.881i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (-5.53 + 1.48i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.258 - 0.0455i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (-0.573 + 6.54i)T + (-42.3 - 7.46i)T^{2} \)
47 \( 1 + (11.9 - 5.56i)T + (30.2 - 36.0i)T^{2} \)
53 \( 1 + (9.15 + 9.15i)T + 53iT^{2} \)
59 \( 1 + (4.98 + 4.18i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (14.0 + 5.13i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (3.34 - 2.34i)T + (22.9 - 62.9i)T^{2} \)
71 \( 1 + (-9.56 - 5.52i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.32 - 0.623i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-2.06 + 0.364i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (5.39 - 7.69i)T + (-28.3 - 77.9i)T^{2} \)
89 \( 1 + (-1.90 - 3.29i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.17 + 0.627i)T + (95.5 + 16.8i)T^{2} \)
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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

−12.24866534377630973784502385920, −11.29896161030175588103045108700, −10.14765608870265169714046018071, −9.383809842216087206251988242754, −8.015841814489051551796170786622, −6.58541700135475499592437965394, −6.10789813974056648795115349318, −4.98354528917355169643562673041, −4.05431294458814606930417169524, −1.81554551080132988763969218777, 1.39677677731163618377617472832, 3.17570090883762488326061285671, 4.68520498058350383825984503296, 5.85446256013743660967492428383, 6.30434028527976866861863285478, 7.61963301760474160442486285135, 9.350813243192121706065566330353, 10.13808201695676951396321565473, 11.02839051113452761582959044866, 11.71090928263355561365852147114

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