Properties

Label 2-18e2-108.31-c2-0-22
Degree $2$
Conductor $324$
Sign $0.997 + 0.0740i$
Analytic cond. $8.82836$
Root an. cond. $2.97125$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.343 + 1.97i)2-s + (−3.76 − 1.35i)4-s + (−0.608 + 3.44i)5-s + (−5.71 − 6.81i)7-s + (3.96 − 6.94i)8-s + (−6.58 − 2.38i)10-s + (−5.38 + 0.949i)11-s + (21.3 − 7.76i)13-s + (15.3 − 8.92i)14-s + (12.3 + 10.1i)16-s + (8.12 − 14.0i)17-s + (−19.5 + 11.2i)19-s + (6.95 − 12.1i)20-s + (−0.0193 − 10.9i)22-s + (22.1 − 26.4i)23-s + ⋯
L(s)  = 1  + (−0.171 + 0.985i)2-s + (−0.940 − 0.338i)4-s + (−0.121 + 0.689i)5-s + (−0.816 − 0.973i)7-s + (0.495 − 0.868i)8-s + (−0.658 − 0.238i)10-s + (−0.489 + 0.0862i)11-s + (1.64 − 0.597i)13-s + (1.09 − 0.637i)14-s + (0.770 + 0.637i)16-s + (0.478 − 0.827i)17-s + (−1.02 + 0.593i)19-s + (0.347 − 0.607i)20-s + (−0.000881 − 0.496i)22-s + (0.963 − 1.14i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0740i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.997 + 0.0740i$
Analytic conductor: \(8.82836\)
Root analytic conductor: \(2.97125\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1),\ 0.997 + 0.0740i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.04244 - 0.0386620i\)
\(L(\frac12)\) \(\approx\) \(1.04244 - 0.0386620i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.343 - 1.97i)T \)
3 \( 1 \)
good5 \( 1 + (0.608 - 3.44i)T + (-23.4 - 8.55i)T^{2} \)
7 \( 1 + (5.71 + 6.81i)T + (-8.50 + 48.2i)T^{2} \)
11 \( 1 + (5.38 - 0.949i)T + (113. - 41.3i)T^{2} \)
13 \( 1 + (-21.3 + 7.76i)T + (129. - 108. i)T^{2} \)
17 \( 1 + (-8.12 + 14.0i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (19.5 - 11.2i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-22.1 + 26.4i)T + (-91.8 - 520. i)T^{2} \)
29 \( 1 + (24.1 + 8.78i)T + (644. + 540. i)T^{2} \)
31 \( 1 + (-14.3 + 17.1i)T + (-166. - 946. i)T^{2} \)
37 \( 1 + (-7.88 + 13.6i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-49.6 + 18.0i)T + (1.28e3 - 1.08e3i)T^{2} \)
43 \( 1 + (-9.61 + 1.69i)T + (1.73e3 - 632. i)T^{2} \)
47 \( 1 + (14.9 + 17.8i)T + (-383. + 2.17e3i)T^{2} \)
53 \( 1 + 4.09T + 2.80e3T^{2} \)
59 \( 1 + (28.0 + 4.93i)T + (3.27e3 + 1.19e3i)T^{2} \)
61 \( 1 + (31.1 - 26.1i)T + (646. - 3.66e3i)T^{2} \)
67 \( 1 + (30.9 + 84.9i)T + (-3.43e3 + 2.88e3i)T^{2} \)
71 \( 1 + (-87.8 - 50.7i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (16.0 + 27.7i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (8.65 - 23.7i)T + (-4.78e3 - 4.01e3i)T^{2} \)
83 \( 1 + (-16.3 + 45.0i)T + (-5.27e3 - 4.42e3i)T^{2} \)
89 \( 1 + (45.0 + 78.0i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (25.7 + 146. i)T + (-8.84e3 + 3.21e3i)T^{2} \)
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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

−10.87697695944903872591529022759, −10.52132421443111011881458167068, −9.443507902525643331702761074834, −8.352307805754796589561807065495, −7.42195804905365616913317435348, −6.60516606477105588181034359192, −5.78981637364024850933050992411, −4.27450947798534789703983510455, −3.23130048232324331625695731350, −0.60306542417761106738647586382, 1.32281001198134341787280350009, 2.87916187677821198177972167044, 3.99925086413413581630996090401, 5.28626919186767726415126644407, 6.33995548237386436768793616601, 8.073243008353877368622529976202, 8.903815604651975322155982541086, 9.343594859058755974373006246080, 10.67445860414737028783495178090, 11.31925072541685184073502131748

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