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Properties

Label 2-18e2-12.11-c3-0-13
Degree $2$
Conductor $324$
Sign $0.205 - 0.978i$
Analytic cond. $19.1166$
Root an. cond. $4.37225$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.19 − 1.78i)2-s + (1.64 − 7.82i)4-s + 11.9i·5-s + 21.2i·7-s + (−10.3 − 20.1i)8-s + (21.2 + 26.1i)10-s − 4.65·11-s − 59.3·13-s + (37.9 + 46.7i)14-s + (−58.5 − 25.7i)16-s + 73.2i·17-s + 85.4i·19-s + (93.2 + 19.6i)20-s + (−10.2 + 8.30i)22-s − 159.·23-s + ⋯
L(s)  = 1  + (0.776 − 0.630i)2-s + (0.205 − 0.978i)4-s + 1.06i·5-s + 1.14i·7-s + (−0.456 − 0.889i)8-s + (0.671 + 0.827i)10-s − 0.127·11-s − 1.26·13-s + (0.724 + 0.892i)14-s + (−0.915 − 0.402i)16-s + 1.04i·17-s + 1.03i·19-s + (1.04 + 0.219i)20-s + (−0.0991 + 0.0804i)22-s − 1.44·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.205 - 0.978i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.205 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.205 - 0.978i$
Analytic conductor: \(19.1166\)
Root analytic conductor: \(4.37225\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :3/2),\ 0.205 - 0.978i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.831441703\)
\(L(\frac12)\) \(\approx\) \(1.831441703\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.19 + 1.78i)T \)
3 \( 1 \)
good5 \( 1 - 11.9iT - 125T^{2} \)
7 \( 1 - 21.2iT - 343T^{2} \)
11 \( 1 + 4.65T + 1.33e3T^{2} \)
13 \( 1 + 59.3T + 2.19e3T^{2} \)
17 \( 1 - 73.2iT - 4.91e3T^{2} \)
19 \( 1 - 85.4iT - 6.85e3T^{2} \)
23 \( 1 + 159.T + 1.21e4T^{2} \)
29 \( 1 + 31.3iT - 2.43e4T^{2} \)
31 \( 1 - 268. iT - 2.97e4T^{2} \)
37 \( 1 - 326.T + 5.06e4T^{2} \)
41 \( 1 + 125. iT - 6.89e4T^{2} \)
43 \( 1 + 95.5iT - 7.95e4T^{2} \)
47 \( 1 - 525.T + 1.03e5T^{2} \)
53 \( 1 + 449. iT - 1.48e5T^{2} \)
59 \( 1 + 180.T + 2.05e5T^{2} \)
61 \( 1 - 624.T + 2.26e5T^{2} \)
67 \( 1 - 887. iT - 3.00e5T^{2} \)
71 \( 1 + 1.08e3T + 3.57e5T^{2} \)
73 \( 1 + 683.T + 3.89e5T^{2} \)
79 \( 1 - 13.7iT - 4.93e5T^{2} \)
83 \( 1 + 863.T + 5.71e5T^{2} \)
89 \( 1 + 540. iT - 7.04e5T^{2} \)
97 \( 1 - 803.T + 9.12e5T^{2} \)
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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

−11.62144333968458534258762580213, −10.38390727709622782265943121089, −10.03165418689939910998096475567, −8.694088169906548518277386410079, −7.34817504107456824380223270440, −6.19473796548051517412005058872, −5.50130703251040594606337308851, −4.09131399412434738482285678882, −2.85281737293536290825137443319, −2.00962402424245254473954177680, 0.47628620065680618580310375441, 2.58813160610825645443701858346, 4.26738715550811181021504339678, 4.73333899369605908126168504923, 5.92463187417357839792710496815, 7.27789142194832376728057973052, 7.72906758269353932882425911137, 8.999549698164449516569131397734, 9.915256515489775980397017906521, 11.29068388488909548028335224968

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