Properties

Label 2-18e2-324.311-c1-0-3
Degree $2$
Conductor $324$
Sign $-0.399 - 0.916i$
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  + (−1.36 − 0.376i)2-s + (−1.73 + 0.0251i)3-s + (1.71 + 1.02i)4-s + (−0.958 + 0.286i)5-s + (2.37 + 0.618i)6-s + (1.31 − 0.569i)7-s + (−1.95 − 2.04i)8-s + (2.99 − 0.0872i)9-s + (1.41 − 0.0298i)10-s + (−2.49 + 0.592i)11-s + (−2.99 − 1.73i)12-s + (−1.52 − 0.999i)13-s + (−2.01 + 0.278i)14-s + (1.65 − 0.521i)15-s + (1.88 + 3.52i)16-s + (0.373 + 1.02i)17-s + ⋯
L(s)  = 1  + (−0.963 − 0.266i)2-s + (−0.999 + 0.0145i)3-s + (0.857 + 0.513i)4-s + (−0.428 + 0.128i)5-s + (0.967 + 0.252i)6-s + (0.498 − 0.215i)7-s + (−0.689 − 0.723i)8-s + (0.999 − 0.0290i)9-s + (0.447 − 0.00944i)10-s + (−0.753 + 0.178i)11-s + (−0.865 − 0.501i)12-s + (−0.421 − 0.277i)13-s + (−0.537 + 0.0744i)14-s + (0.426 − 0.134i)15-s + (0.472 + 0.881i)16-s + (0.0905 + 0.248i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.399 - 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.399 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.399 - 0.916i$
Analytic conductor: \(2.58715\)
Root analytic conductor: \(1.60846\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1/2),\ -0.399 - 0.916i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.150912 + 0.230308i\)
\(L(\frac12)\) \(\approx\) \(0.150912 + 0.230308i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 + 0.376i)T \)
3 \( 1 + (1.73 - 0.0251i)T \)
good5 \( 1 + (0.958 - 0.286i)T + (4.17 - 2.74i)T^{2} \)
7 \( 1 + (-1.31 + 0.569i)T + (4.80 - 5.09i)T^{2} \)
11 \( 1 + (2.49 - 0.592i)T + (9.82 - 4.93i)T^{2} \)
13 \( 1 + (1.52 + 0.999i)T + (5.14 + 11.9i)T^{2} \)
17 \( 1 + (-0.373 - 1.02i)T + (-13.0 + 10.9i)T^{2} \)
19 \( 1 + (0.642 - 1.76i)T + (-14.5 - 12.2i)T^{2} \)
23 \( 1 + (2.94 - 6.83i)T + (-15.7 - 16.7i)T^{2} \)
29 \( 1 + (1.06 - 0.0622i)T + (28.8 - 3.36i)T^{2} \)
31 \( 1 + (-1.23 - 10.5i)T + (-30.1 + 7.14i)T^{2} \)
37 \( 1 + (4.08 + 3.42i)T + (6.42 + 36.4i)T^{2} \)
41 \( 1 + (-2.74 - 5.46i)T + (-24.4 + 32.8i)T^{2} \)
43 \( 1 + (-1.81 + 1.71i)T + (2.50 - 42.9i)T^{2} \)
47 \( 1 + (2.78 + 0.325i)T + (45.7 + 10.8i)T^{2} \)
53 \( 1 + (1.17 + 0.680i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.60 + 1.09i)T + (52.7 + 26.4i)T^{2} \)
61 \( 1 + (4.28 - 5.75i)T + (-17.4 - 58.4i)T^{2} \)
67 \( 1 + (0.322 + 0.0187i)T + (66.5 + 7.77i)T^{2} \)
71 \( 1 + (-1.36 + 7.74i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (1.92 + 10.8i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (-4.89 + 9.75i)T + (-47.1 - 63.3i)T^{2} \)
83 \( 1 + (-0.440 - 0.221i)T + (49.5 + 66.5i)T^{2} \)
89 \( 1 + (-13.1 + 2.31i)T + (83.6 - 30.4i)T^{2} \)
97 \( 1 + (2.40 - 8.01i)T + (-81.0 - 53.3i)T^{2} \)
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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

−11.77588142439411295899354954246, −10.80436182453048204191903922361, −10.32580155484472697752203377786, −9.296926501100551763024765094687, −7.83272411858788734734233918281, −7.48273921372743127754068969789, −6.19407775308391882494107950253, −5.01821027625017030993142697672, −3.53481994228327027328506994532, −1.64015708525214845350182967991, 0.29696835935422404908628847037, 2.23454160676716889755934570681, 4.47210511710824828555133461182, 5.56006717622267793645482355944, 6.53605229614677941791205498524, 7.60816937337947722330432500497, 8.315432437813494055651330364762, 9.591277512755338850150297543121, 10.40796340854862326971438951422, 11.27057400326623157938272755269

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