Properties

Label 2-18e2-9.4-c1-0-1
Degree $2$
Conductor $324$
Sign $0.766 - 0.642i$
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  + (2 + 3.46i)7-s + (−1 + 1.73i)13-s + 8·19-s + (2.5 + 4.33i)25-s + (2 − 3.46i)31-s − 10·37-s + (−4 − 6.92i)43-s + (−4.49 + 7.79i)49-s + (−7 − 12.1i)61-s + (8 − 13.8i)67-s − 10·73-s + (2 + 3.46i)79-s − 7.99·91-s + (−7 − 12.1i)97-s + (−10 + 17.3i)103-s + ⋯
L(s)  = 1  + (0.755 + 1.30i)7-s + (−0.277 + 0.480i)13-s + 1.83·19-s + (0.5 + 0.866i)25-s + (0.359 − 0.622i)31-s − 1.64·37-s + (−0.609 − 1.05i)43-s + (−0.642 + 1.11i)49-s + (−0.896 − 1.55i)61-s + (0.977 − 1.69i)67-s − 1.17·73-s + (0.225 + 0.389i)79-s − 0.838·91-s + (−0.710 − 1.23i)97-s + (−0.985 + 1.70i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.766 - 0.642i$
Analytic conductor: \(2.58715\)
Root analytic conductor: \(1.60846\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1/2),\ 0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31762 + 0.479574i\)
\(L(\frac12)\) \(\approx\) \(1.31762 + 0.479574i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-2 - 3.46i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 8T + 19T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 10T + 37T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4 + 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7 + 12.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8 + 13.8i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 10T + 73T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (7 + 12.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.82235422542429105925070832049, −10.98076526105920228072948946830, −9.671473231001205570019232826521, −8.961855264884587509955799701775, −7.997507265526095423109297164313, −6.95141502013665706448618407516, −5.58855634823398388109491201333, −4.92950719856807186528671678599, −3.24801769331644665393840249736, −1.83735858653833911855574313271, 1.18259588849665885008193160449, 3.14422391153317343904853441402, 4.44450720019173676674692811510, 5.39649456725599843144128363972, 6.91031273329751123854354002521, 7.61686246832907793041088019723, 8.557297404849904883139388388967, 9.892946035630855793755962860148, 10.51337003153214371961132201644, 11.46958622415999571144457931992

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