Properties

Label 2-18e2-324.263-c1-0-14
Degree $2$
Conductor $324$
Sign $0.207 - 0.978i$
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  + (−0.237 + 1.39i)2-s + (−1.41 − 1.00i)3-s + (−1.88 − 0.661i)4-s + (−0.359 + 0.715i)5-s + (1.73 − 1.73i)6-s + (1.97 − 0.592i)7-s + (1.37 − 2.47i)8-s + (0.988 + 2.83i)9-s + (−0.911 − 0.670i)10-s + (−0.0389 + 0.669i)11-s + (2.00 + 2.82i)12-s + (0.454 − 0.610i)13-s + (0.356 + 2.90i)14-s + (1.22 − 0.649i)15-s + (3.12 + 2.49i)16-s + (4.07 + 4.85i)17-s + ⋯
L(s)  = 1  + (−0.167 + 0.985i)2-s + (−0.815 − 0.578i)3-s + (−0.943 − 0.330i)4-s + (−0.160 + 0.319i)5-s + (0.707 − 0.706i)6-s + (0.748 − 0.224i)7-s + (0.484 − 0.874i)8-s + (0.329 + 0.944i)9-s + (−0.288 − 0.212i)10-s + (−0.0117 + 0.201i)11-s + (0.577 + 0.816i)12-s + (0.125 − 0.169i)13-s + (0.0952 + 0.775i)14-s + (0.316 − 0.167i)15-s + (0.780 + 0.624i)16-s + (0.987 + 1.17i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.207 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.207 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.207 - 0.978i$
Analytic conductor: \(2.58715\)
Root analytic conductor: \(1.60846\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1/2),\ 0.207 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.684196 + 0.554461i\)
\(L(\frac12)\) \(\approx\) \(0.684196 + 0.554461i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.237 - 1.39i)T \)
3 \( 1 + (1.41 + 1.00i)T \)
good5 \( 1 + (0.359 - 0.715i)T + (-2.98 - 4.01i)T^{2} \)
7 \( 1 + (-1.97 + 0.592i)T + (5.84 - 3.84i)T^{2} \)
11 \( 1 + (0.0389 - 0.669i)T + (-10.9 - 1.27i)T^{2} \)
13 \( 1 + (-0.454 + 0.610i)T + (-3.72 - 12.4i)T^{2} \)
17 \( 1 + (-4.07 - 4.85i)T + (-2.95 + 16.7i)T^{2} \)
19 \( 1 + (-0.317 + 0.378i)T + (-3.29 - 18.7i)T^{2} \)
23 \( 1 + (2.67 - 8.95i)T + (-19.2 - 12.6i)T^{2} \)
29 \( 1 + (-6.37 - 2.75i)T + (19.9 + 21.0i)T^{2} \)
31 \( 1 + (-1.64 + 1.54i)T + (1.80 - 30.9i)T^{2} \)
37 \( 1 + (0.869 + 4.93i)T + (-34.7 + 12.6i)T^{2} \)
41 \( 1 + (-1.06 + 9.11i)T + (-39.8 - 9.45i)T^{2} \)
43 \( 1 + (-3.62 + 5.50i)T + (-17.0 - 39.4i)T^{2} \)
47 \( 1 + (-1.33 + 1.41i)T + (-2.73 - 46.9i)T^{2} \)
53 \( 1 + (-3.66 + 2.11i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.449 - 7.71i)T + (-58.6 + 6.84i)T^{2} \)
61 \( 1 + (-1.86 - 0.442i)T + (54.5 + 27.3i)T^{2} \)
67 \( 1 + (4.44 - 1.91i)T + (45.9 - 48.7i)T^{2} \)
71 \( 1 + (-1.25 - 0.457i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (9.76 - 3.55i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (-0.220 - 1.88i)T + (-76.8 + 18.2i)T^{2} \)
83 \( 1 + (7.03 - 0.822i)T + (80.7 - 19.1i)T^{2} \)
89 \( 1 + (-1.95 - 5.36i)T + (-68.1 + 57.2i)T^{2} \)
97 \( 1 + (-10.1 + 5.11i)T + (57.9 - 77.8i)T^{2} \)
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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

−11.87038062535991222860693755618, −10.77418326019417048282481827503, −10.09445892473834212488784261270, −8.669242867061323587605212261597, −7.65740008650263589468767765944, −7.18529294927739139943378295055, −5.91475194377883602671087123542, −5.25873753047477231057340327019, −3.92530747165123055178825423432, −1.37000806934156910660660227713, 0.909586118667467160239578381323, 2.89065861451257222553709251929, 4.46299132207509198788007842916, 4.93181155184597588901188237971, 6.30218433763002969450560724136, 7.965561809231903473190083402852, 8.783805285930114709895818763869, 9.879223595529143043702802672814, 10.50438394222154843996057288794, 11.55853843277611669769663338954

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