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Properties

Label 2-270-45.2-c1-0-5
Degree $2$
Conductor $270$
Sign $-0.974 - 0.223i$
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  + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s + (−1.36 − 1.76i)5-s + (−2.56 + 0.686i)7-s + (0.707 + 0.707i)8-s + (−1.35 + 1.77i)10-s + (−4.15 − 2.39i)11-s + (0.581 + 0.155i)13-s + (1.32 + 2.29i)14-s + (0.500 − 0.866i)16-s + (−4.40 + 4.40i)17-s − 5.19i·19-s + (2.06 + 0.847i)20-s + (−1.24 + 4.63i)22-s + (0.681 − 2.54i)23-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (−0.433 + 0.249i)4-s + (−0.612 − 0.790i)5-s + (−0.968 + 0.259i)7-s + (0.249 + 0.249i)8-s + (−0.428 + 0.562i)10-s + (−1.25 − 0.723i)11-s + (0.161 + 0.0432i)13-s + (0.354 + 0.613i)14-s + (0.125 − 0.216i)16-s + (−1.06 + 1.06i)17-s − 1.19i·19-s + (0.462 + 0.189i)20-s + (−0.264 + 0.988i)22-s + (0.142 − 0.530i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 - 0.223i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.974 - 0.223i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(270\)    =    \(2 \cdot 3^{3} \cdot 5\)
Sign: $-0.974 - 0.223i$
Analytic conductor: \(2.15596\)
Root analytic conductor: \(1.46831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{270} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 270,\ (\ :1/2),\ -0.974 - 0.223i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0402721 + 0.355963i\)
\(L(\frac12)\) \(\approx\) \(0.0402721 + 0.355963i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
3 \( 1 \)
5 \( 1 + (1.36 + 1.76i)T \)
good7 \( 1 + (2.56 - 0.686i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (4.15 + 2.39i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.581 - 0.155i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (4.40 - 4.40i)T - 17iT^{2} \)
19 \( 1 + 5.19iT - 19T^{2} \)
23 \( 1 + (-0.681 + 2.54i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-0.920 + 1.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.03 + 3.53i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.632 - 0.632i)T + 37iT^{2} \)
41 \( 1 + (-5.58 + 3.22i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.644 + 2.40i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-1.02 - 3.82i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.31 - 1.31i)T + 53iT^{2} \)
59 \( 1 + (-0.0645 - 0.111i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.27 + 10.8i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.85 + 10.6i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 10.4iT - 71T^{2} \)
73 \( 1 + (-3.30 + 3.30i)T - 73iT^{2} \)
79 \( 1 + (-3.62 - 2.09i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (11.1 - 2.97i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + 2.04T + 89T^{2} \)
97 \( 1 + (16.7 - 4.47i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.26901122104197943662724011268, −10.68761343698990942571194785610, −9.411983300583221759056561928183, −8.692102687829408181352324275766, −7.82348914314882094992906969201, −6.33936285730757336198855551391, −5.04050082430138537416179103989, −3.84120360089513925311071488096, −2.55328551077403687300516082791, −0.27703363464328440497450702590, 2.80327254234914132779514206153, 4.14416145507245689825747942530, 5.52205355921069839169796078788, 6.80683007320636637377148601998, 7.33555453746422285654188455024, 8.376693867762161651522176309246, 9.692539423951949943609452164004, 10.33762482469479760512718018297, 11.35674518736751135493858123525, 12.60023433471064267081782318417

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