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Properties

Label 2-18e2-108.47-c1-0-10
Degree $2$
Conductor $324$
Sign $0.888 - 0.459i$
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  + (1.30 + 0.540i)2-s + (1.41 + 1.41i)4-s + (−0.137 + 0.0242i)5-s + (2.98 − 3.56i)7-s + (1.08 + 2.61i)8-s + (−0.192 − 0.0424i)10-s + (−0.182 + 1.03i)11-s + (−1.46 − 0.532i)13-s + (5.82 − 3.04i)14-s + (0.0146 + 3.99i)16-s + (−5.25 + 3.03i)17-s + (3.80 + 2.19i)19-s + (−0.228 − 0.159i)20-s + (−0.797 + 1.25i)22-s + (1.94 − 1.62i)23-s + ⋯
L(s)  = 1  + (0.924 + 0.381i)2-s + (0.708 + 0.705i)4-s + (−0.0613 + 0.0108i)5-s + (1.12 − 1.34i)7-s + (0.385 + 0.922i)8-s + (−0.0608 − 0.0134i)10-s + (−0.0550 + 0.312i)11-s + (−0.405 − 0.147i)13-s + (1.55 − 0.812i)14-s + (0.00365 + 0.999i)16-s + (−1.27 + 0.736i)17-s + (0.871 + 0.503i)19-s + (−0.0511 − 0.0356i)20-s + (−0.170 + 0.267i)22-s + (0.404 − 0.339i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.459i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 - 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.888 - 0.459i$
Analytic conductor: \(2.58715\)
Root analytic conductor: \(1.60846\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1/2),\ 0.888 - 0.459i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.28054 + 0.555024i\)
\(L(\frac12)\) \(\approx\) \(2.28054 + 0.555024i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.30 - 0.540i)T \)
3 \( 1 \)
good5 \( 1 + (0.137 - 0.0242i)T + (4.69 - 1.71i)T^{2} \)
7 \( 1 + (-2.98 + 3.56i)T + (-1.21 - 6.89i)T^{2} \)
11 \( 1 + (0.182 - 1.03i)T + (-10.3 - 3.76i)T^{2} \)
13 \( 1 + (1.46 + 0.532i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (5.25 - 3.03i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.80 - 2.19i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.94 + 1.62i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (0.210 + 0.578i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (3.18 + 3.79i)T + (-5.38 + 30.5i)T^{2} \)
37 \( 1 + (1.43 + 2.49i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.25 + 6.18i)T + (-31.4 - 26.3i)T^{2} \)
43 \( 1 + (2.15 + 0.380i)T + (40.4 + 14.7i)T^{2} \)
47 \( 1 + (8.46 + 7.10i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 - 8.73iT - 53T^{2} \)
59 \( 1 + (1.42 + 8.08i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-8.56 - 7.18i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (1.93 - 5.30i)T + (-51.3 - 43.0i)T^{2} \)
71 \( 1 + (-2.33 - 4.04i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.64 - 9.77i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.572 + 1.57i)T + (-60.5 + 50.7i)T^{2} \)
83 \( 1 + (-2.20 + 0.802i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (7.63 + 4.40i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.978 - 5.54i)T + (-91.1 - 33.1i)T^{2} \)
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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

−11.63087243200284266115756188943, −11.03654739370892328465699265614, −10.10021879997494107196766660338, −8.504660894020844154729359129090, −7.57321750225288653385049263569, −6.99138588753817636351511675565, −5.59559661293559171071809599988, −4.53749185838633822391709639638, −3.77625221704396787637080845092, −1.93951885262832624383315783088, 1.89577938394080385286196305109, 3.01078684976204306334847028698, 4.69917905146511948920961875416, 5.26040105635618606993226663328, 6.42078714427008554769093488602, 7.62873034140634716706642196296, 8.833197000949349475816566330170, 9.730006983503048381497961906549, 11.27601756576479608967854266476, 11.40693973391351022929657959524

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