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Properties

Label 2-18e2-36.11-c1-0-13
Degree $2$
Conductor $324$
Sign $0.995 - 0.0962i$
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  + (1.40 − 0.190i)2-s + (1.92 − 0.535i)4-s + (−1.93 + 1.11i)5-s + (3.35 + 1.93i)7-s + (2.59 − 1.11i)8-s + (−2.5 + 1.93i)10-s + (0.866 − 1.5i)11-s + (−1 − 1.73i)13-s + (5.06 + 2.07i)14-s + (3.42 − 2.06i)16-s + 4.47i·17-s + (−3.13 + 3.19i)20-s + (0.927 − 2.26i)22-s + (−3.46 − 6i)23-s + (−1.73 − 2.23i)26-s + ⋯
L(s)  = 1  + (0.990 − 0.135i)2-s + (0.963 − 0.267i)4-s + (−0.866 + 0.499i)5-s + (1.26 + 0.731i)7-s + (0.918 − 0.395i)8-s + (−0.790 + 0.612i)10-s + (0.261 − 0.452i)11-s + (−0.277 − 0.480i)13-s + (1.35 + 0.554i)14-s + (0.856 − 0.515i)16-s + 1.08i·17-s + (−0.700 + 0.713i)20-s + (0.197 − 0.483i)22-s + (−0.722 − 1.25i)23-s + (−0.339 − 0.438i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.995 - 0.0962i$
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: \((2,\ 324,\ (\ :1/2),\ 0.995 - 0.0962i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.31972 + 0.111880i\)
\(L(\frac12)\) \(\approx\) \(2.31972 + 0.111880i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 + 0.190i)T \)
3 \( 1 \)
good5 \( 1 + (1.93 - 1.11i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-3.35 - 1.93i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.866 + 1.5i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 4.47iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + (3.46 + 6i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.87 + 2.23i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.35 + 1.93i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + (7.74 - 4.47i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.70 + 3.87i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.73 - 3i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 2.23iT - 53T^{2} \)
59 \( 1 + (-1.73 - 3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.70 - 3.87i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 - 5T + 73T^{2} \)
79 \( 1 + (6.70 + 3.87i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.06 + 10.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 4.47iT - 89T^{2} \)
97 \( 1 + (5.5 - 9.52i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.71482645846357875736306222338, −11.05407649238278583506728258253, −10.19286715121765868837104211435, −8.414737905885622513449785658375, −7.85978927200828231734474819616, −6.59639578162241166160810658971, −5.55417691542885771562505030011, −4.48501086374034286918113710401, −3.41479695010267861652246598755, −2.00341630920871864525283406034, 1.72321818550642448142478171157, 3.64059817601968193976126765481, 4.56506110634631070314352158379, 5.23150967764624715764226567605, 6.91929460858255623645526267337, 7.58455301101970663035704578016, 8.419021485259403455061332597214, 9.913720622529148138487074554794, 11.15979262321500357876891580537, 11.72368846622362158605033703884

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