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Properties

Label 2-18e2-27.13-c1-0-0
Degree $2$
Conductor $324$
Sign $0.962 - 0.271i$
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  + (0.761 + 0.639i)5-s + (1.35 + 0.492i)7-s + (2.56 − 2.15i)11-s + (−0.337 + 1.91i)13-s + (−1.16 + 2.01i)17-s + (3.38 + 5.86i)19-s + (8.41 − 3.06i)23-s + (−0.696 − 3.95i)25-s + (−0.847 − 4.80i)29-s + (−5.81 + 2.11i)31-s + (0.715 + 1.23i)35-s + (−0.0829 + 0.143i)37-s + (−1.87 + 10.6i)41-s + (6.82 − 5.73i)43-s + (−6.14 − 2.23i)47-s + ⋯
L(s)  = 1  + (0.340 + 0.285i)5-s + (0.511 + 0.186i)7-s + (0.773 − 0.648i)11-s + (−0.0936 + 0.531i)13-s + (−0.281 + 0.487i)17-s + (0.776 + 1.34i)19-s + (1.75 − 0.638i)23-s + (−0.139 − 0.790i)25-s + (−0.157 − 0.892i)29-s + (−1.04 + 0.380i)31-s + (0.120 + 0.209i)35-s + (−0.0136 + 0.0236i)37-s + (−0.292 + 1.65i)41-s + (1.04 − 0.873i)43-s + (−0.896 − 0.326i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.271i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 - 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.962 - 0.271i$
Analytic conductor: \(2.58715\)
Root analytic conductor: \(1.60846\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1/2),\ 0.962 - 0.271i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48614 + 0.205528i\)
\(L(\frac12)\) \(\approx\) \(1.48614 + 0.205528i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.761 - 0.639i)T + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (-1.35 - 0.492i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (-2.56 + 2.15i)T + (1.91 - 10.8i)T^{2} \)
13 \( 1 + (0.337 - 1.91i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (1.16 - 2.01i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.38 - 5.86i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-8.41 + 3.06i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (0.847 + 4.80i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (5.81 - 2.11i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (0.0829 - 0.143i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.87 - 10.6i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-6.82 + 5.73i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (6.14 + 2.23i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + 10.7T + 53T^{2} \)
59 \( 1 + (3.02 + 2.53i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (7.91 + 2.88i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (0.00383 - 0.0217i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (4.53 - 7.85i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.26 + 3.92i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.00 + 5.70i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (0.898 + 5.09i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (1.91 + 3.32i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.40 - 4.53i)T + (16.8 - 95.5i)T^{2} \)
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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

−11.55482080943267739985330711561, −10.84915536162560870027819027242, −9.760977701502983386457291846394, −8.863398984401998413261425581524, −7.934560675085914490751516427460, −6.69757977412104833643244553459, −5.85554454930242673527504758354, −4.58108638373919107024908690591, −3.26395553216453102734228376226, −1.64714001617344541725863502267, 1.41068864353297758443608063249, 3.12025520557214309795515443684, 4.67050985617409318556925085012, 5.42905149547584584787917246117, 6.94057683305685778644296476657, 7.56920355116155918558461518540, 9.195357729871180667938746443962, 9.303416367198532912608659290325, 10.87065894845633804293402704815, 11.39226149950343122242688903813

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