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Properties

Label 2-18e2-36.11-c1-0-9
Degree $2$
Conductor $324$
Sign $0.642 - 0.766i$
Analytic cond. $2.58715$
Root an. cond. $1.60846$
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  + 1.41i·2-s − 2.00·4-s + (2.44 − 1.41i)5-s + (1.5 + 0.866i)7-s − 2.82i·8-s + (2.00 + 3.46i)10-s + (2.44 − 4.24i)11-s + (0.5 + 0.866i)13-s + (−1.22 + 2.12i)14-s + 4.00·16-s + 2.82i·17-s + 5.19i·19-s + (−4.89 + 2.82i)20-s + (6 + 3.46i)22-s + (−2.44 − 4.24i)23-s + ⋯
L(s)  = 1  + 0.999i·2-s − 1.00·4-s + (1.09 − 0.632i)5-s + (0.566 + 0.327i)7-s − 1.00i·8-s + (0.632 + 1.09i)10-s + (0.738 − 1.27i)11-s + (0.138 + 0.240i)13-s + (−0.327 + 0.566i)14-s + 1.00·16-s + 0.685i·17-s + 1.19i·19-s + (−1.09 + 0.632i)20-s + (1.27 + 0.738i)22-s + (−0.510 − 0.884i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.642 - 0.766i$
Analytic conductor: \(2.58715\)
Root analytic conductor: \(1.60846\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1/2),\ 0.642 - 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37247 + 0.639996i\)
\(L(\frac12)\) \(\approx\) \(1.37247 + 0.639996i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
3 \( 1 \)
good5 \( 1 + (-2.44 + 1.41i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.5 - 0.866i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.44 + 4.24i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 2.82iT - 17T^{2} \)
19 \( 1 - 5.19iT - 19T^{2} \)
23 \( 1 + (2.44 + 4.24i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.89 - 2.82i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (3 - 1.73i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + (4.89 - 2.82i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (3 + 1.73i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.44 + 4.24i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 5.65iT - 53T^{2} \)
59 \( 1 + (2.44 + 4.24i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.5 - 9.52i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.5 + 6.06i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + T + 73T^{2} \)
79 \( 1 + (-1.5 - 0.866i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.89 - 8.48i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 2.82iT - 89T^{2} \)
97 \( 1 + (-6.5 + 11.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.96057262449977929820904440380, −10.57289360446221481609671414113, −9.631260338829108254848942110011, −8.571327295163973031235167366656, −8.333831959649029851067196926577, −6.61886577976753218685881837302, −5.89657157271422704362469481788, −5.09912967013019240860000542272, −3.75136778458233557217615719392, −1.49632007829764188428102868808, 1.61397288815048488848598089754, 2.71937789360993006903007416664, 4.25396527917363217207197367179, 5.26394759092418200156097167610, 6.60539158364233464493275247096, 7.72721308057389181645785926366, 9.166634749533045573289977044923, 9.734582322216926957970529098849, 10.54426791488672475410968660049, 11.41832594886181949102295021499

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