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Properties

Label 2-18e2-36.23-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: $\chi_{324} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1/2),\ 0.342 - 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.87334 + 1.31173i\)
\(L(\frac12)\) \(\approx\) \(1.87334 + 1.31173i\)
\(L(\frac{3}{2})\) 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

−12.02262797909805494857632927104, −10.84640598747523223453664067194, −10.13691898755137995352601237285, −8.829380757773911472741079366421, −7.85980688529579136682620408308, −6.49871807797037114174577578359, −6.03085515680400797077667810690, −5.06969229836237586808339759571, −3.33504661366870448580889694333, −2.52528001744762015274435002546, 1.58443480156855741994996601698, 2.86794989817528221003860445195, 4.46236890707131006547338144317, 5.25024556354288219746195837122, 6.33571767600225429320114975449, 7.26772178380717655530007967521, 8.976146107696248300656337058446, 9.869563968441995294874045596917, 10.37627739790582095440643510987, 11.58163146759946696868396345960

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