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Properties

Label 2-18e2-12.11-c1-0-11
Degree $2$
Conductor $324$
Sign $0.933 - 0.359i$
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  + (1.39 − 0.258i)2-s + (1.86 − 0.719i)4-s + 1.93i·5-s + 3.93i·7-s + (2.40 − 1.48i)8-s + (0.499 + 2.68i)10-s − 2.03·11-s + 2.46·13-s + (1.01 + 5.46i)14-s + (2.96 − 2.68i)16-s − 4.76i·17-s − 6.81i·19-s + (1.39 + 3.60i)20-s + (−2.83 + 0.526i)22-s − 7.59·23-s + ⋯
L(s)  = 1  + (0.983 − 0.183i)2-s + (0.933 − 0.359i)4-s + 0.863i·5-s + 1.48i·7-s + (0.851 − 0.524i)8-s + (0.158 + 0.849i)10-s − 0.613·11-s + 0.683·13-s + (0.272 + 1.46i)14-s + (0.741 − 0.671i)16-s − 1.15i·17-s − 1.56i·19-s + (0.310 + 0.806i)20-s + (−0.603 + 0.112i)22-s − 1.58·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 - 0.359i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.933 - 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.933 - 0.359i$
Analytic conductor: \(2.58715\)
Root analytic conductor: \(1.60846\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1/2),\ 0.933 - 0.359i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.29423 + 0.427087i\)
\(L(\frac12)\) \(\approx\) \(2.29423 + 0.427087i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 + 0.258i)T \)
3 \( 1 \)
good5 \( 1 - 1.93iT - 5T^{2} \)
7 \( 1 - 3.93iT - 7T^{2} \)
11 \( 1 + 2.03T + 11T^{2} \)
13 \( 1 - 2.46T + 13T^{2} \)
17 \( 1 + 4.76iT - 17T^{2} \)
19 \( 1 + 6.81iT - 19T^{2} \)
23 \( 1 + 7.59T + 23T^{2} \)
29 \( 1 + 2.31iT - 29T^{2} \)
31 \( 1 - 2.87iT - 31T^{2} \)
37 \( 1 - 3.73T + 37T^{2} \)
41 \( 1 + 3.20iT - 41T^{2} \)
43 \( 1 - 3.93iT - 43T^{2} \)
47 \( 1 + 5.56T + 47T^{2} \)
53 \( 1 - 10.1iT - 53T^{2} \)
59 \( 1 - 5.56T + 59T^{2} \)
61 \( 1 + 5.73T + 61T^{2} \)
67 \( 1 + 3.93iT - 67T^{2} \)
71 \( 1 - 3.52T + 71T^{2} \)
73 \( 1 + 12.6T + 73T^{2} \)
79 \( 1 + 1.05iT - 79T^{2} \)
83 \( 1 - 4.07T + 83T^{2} \)
89 \( 1 + 3.62iT - 89T^{2} \)
97 \( 1 + 7.46T + 97T^{2} \)
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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

−11.64586303848795579899045198151, −11.10336237079935024296109472322, −10.03908973422784500069204398879, −8.920010171172303318973894569227, −7.61657583487717925175264520773, −6.52852093373649394968285942742, −5.72418243509431627656158375905, −4.68798976197116905751440568680, −3.05882932880258372018172504584, −2.37696786619257676036627068993, 1.58304293381580849473353851010, 3.67116412474480569795861328867, 4.29182870042305271009160013761, 5.55818118587321540651046883018, 6.49275133996948698971627508840, 7.80718389922321831944517615346, 8.274224978202138874109448383783, 10.08333964449917309926936583147, 10.64400734291183526163486529266, 11.75915260610245094299491130493

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