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Properties

Label 2-18e2-324.83-c1-0-32
Degree $2$
Conductor $324$
Sign $0.940 - 0.339i$
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  + (−0.168 + 1.40i)2-s + (1.51 − 0.830i)3-s + (−1.94 − 0.473i)4-s + (−0.117 − 0.497i)5-s + (0.910 + 2.27i)6-s + (0.633 − 0.471i)7-s + (0.992 − 2.64i)8-s + (1.61 − 2.52i)9-s + (0.718 − 0.0817i)10-s + (2.23 − 2.36i)11-s + (−3.34 + 0.895i)12-s + (4.23 + 2.12i)13-s + (0.555 + 0.968i)14-s + (−0.592 − 0.658i)15-s + (3.55 + 1.83i)16-s + (−2.31 + 6.36i)17-s + ⋯
L(s)  = 1  + (−0.119 + 0.992i)2-s + (0.877 − 0.479i)3-s + (−0.971 − 0.236i)4-s + (−0.0527 − 0.222i)5-s + (0.371 + 0.928i)6-s + (0.239 − 0.178i)7-s + (0.350 − 0.936i)8-s + (0.539 − 0.841i)9-s + (0.227 − 0.0258i)10-s + (0.673 − 0.713i)11-s + (−0.966 + 0.258i)12-s + (1.17 + 0.590i)13-s + (0.148 + 0.258i)14-s + (−0.153 − 0.169i)15-s + (0.887 + 0.459i)16-s + (−0.561 + 1.54i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 - 0.339i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.940 - 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.940 - 0.339i$
Analytic conductor: \(2.58715\)
Root analytic conductor: \(1.60846\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1/2),\ 0.940 - 0.339i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55767 + 0.272902i\)
\(L(\frac12)\) \(\approx\) \(1.55767 + 0.272902i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.168 - 1.40i)T \)
3 \( 1 + (-1.51 + 0.830i)T \)
good5 \( 1 + (0.117 + 0.497i)T + (-4.46 + 2.24i)T^{2} \)
7 \( 1 + (-0.633 + 0.471i)T + (2.00 - 6.70i)T^{2} \)
11 \( 1 + (-2.23 + 2.36i)T + (-0.639 - 10.9i)T^{2} \)
13 \( 1 + (-4.23 - 2.12i)T + (7.76 + 10.4i)T^{2} \)
17 \( 1 + (2.31 - 6.36i)T + (-13.0 - 10.9i)T^{2} \)
19 \( 1 + (2.18 + 5.99i)T + (-14.5 + 12.2i)T^{2} \)
23 \( 1 + (3.12 - 4.19i)T + (-6.59 - 22.0i)T^{2} \)
29 \( 1 + (-0.742 + 1.12i)T + (-11.4 - 26.6i)T^{2} \)
31 \( 1 + (-6.37 + 2.75i)T + (21.2 - 22.5i)T^{2} \)
37 \( 1 + (6.15 - 5.16i)T + (6.42 - 36.4i)T^{2} \)
41 \( 1 + (8.56 - 0.498i)T + (40.7 - 4.75i)T^{2} \)
43 \( 1 + (5.08 - 1.52i)T + (35.9 - 23.6i)T^{2} \)
47 \( 1 + (0.840 - 1.94i)T + (-32.2 - 34.1i)T^{2} \)
53 \( 1 + (3.73 - 2.15i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.53 - 1.62i)T + (-3.43 + 58.9i)T^{2} \)
61 \( 1 + (-7.35 + 0.860i)T + (59.3 - 14.0i)T^{2} \)
67 \( 1 + (5.37 + 8.16i)T + (-26.5 + 61.5i)T^{2} \)
71 \( 1 + (0.383 + 2.17i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (-0.885 + 5.02i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (-0.707 - 0.0412i)T + (78.4 + 9.17i)T^{2} \)
83 \( 1 + (0.517 - 8.88i)T + (-82.4 - 9.63i)T^{2} \)
89 \( 1 + (5.10 + 0.900i)T + (83.6 + 30.4i)T^{2} \)
97 \( 1 + (3.30 + 0.783i)T + (86.6 + 43.5i)T^{2} \)
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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

−11.77666304390933627070914530752, −10.56703413688706849772742214256, −9.299905986249888181702274419593, −8.536584572397529071670077026814, −8.155733606512132644943779749673, −6.66075110017239139090495862024, −6.30336205258467262261877387899, −4.52677543334569063606786476598, −3.57857683609170821881923290810, −1.39267461477103309956596373028, 1.79175077142371794341457103029, 3.10641821470262999878868603119, 4.08748899475005138230103536716, 5.14329168781943381602871611752, 6.92393999266255614321156094204, 8.338649169564110941409697371854, 8.728386011653294642494124190908, 9.909072194582034141798669884809, 10.44369953251835332543737671278, 11.48393461006063669596640614501

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