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Properties

Label 2-18e2-36.11-c1-0-10
Degree $2$
Conductor $324$
Sign $0.342 + 0.939i$
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  + (−1.22 + 0.707i)2-s + (0.999 − 1.73i)4-s + (−2.44 + 1.41i)5-s + (−1.5 − 0.866i)7-s + 2.82i·8-s + (1.99 − 3.46i)10-s + (2.44 − 4.24i)11-s + (0.5 + 0.866i)13-s + 2.44·14-s + (−2.00 − 3.46i)16-s − 2.82i·17-s − 5.19i·19-s + 5.65i·20-s + 6.92i·22-s + (−2.44 − 4.24i)23-s + ⋯
L(s)  = 1  + (−0.866 + 0.499i)2-s + (0.499 − 0.866i)4-s + (−1.09 + 0.632i)5-s + (−0.566 − 0.327i)7-s + 0.999i·8-s + (0.632 − 1.09i)10-s + (0.738 − 1.27i)11-s + (0.138 + 0.240i)13-s + 0.654·14-s + (−0.500 − 0.866i)16-s − 0.685i·17-s − 1.19i·19-s + 1.26i·20-s + 1.47i·22-s + (−0.510 − 0.884i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.342 + 0.939i$
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: \((2,\ 324,\ (\ :1/2),\ 0.342 + 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.394274 - 0.276073i\)
\(L(\frac12)\) \(\approx\) \(0.394274 - 0.276073i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 - 0.707i)T \)
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

−11.29590580015308143803140230585, −10.56509422090054354436415682461, −9.376365904650975442616030787927, −8.607503621460782449718296118875, −7.54697307992944941050436848352, −6.81860681185601038923189947835, −5.92466029761046441513729123120, −4.19844623681412881947878476658, −2.88611052024228313022310259255, −0.46586940799749692397480439633, 1.62713609108648205787100399277, 3.46857718759297321924094406682, 4.33234462581163810232217985443, 6.13904444905258672662918632568, 7.40717791477830515842155558723, 8.059651741492884359333396808855, 9.101975160900623508751713420915, 9.793289698365595241125676606392, 10.83514444275068475779660875203, 12.01126620412463733447554604782

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