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Properties

Label 2-18e2-27.22-c1-0-0
Degree $2$
Conductor $324$
Sign $0.946 - 0.321i$
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  + (−0.0952 + 0.0346i)5-s + (0.165 + 0.935i)7-s + (3.80 + 1.38i)11-s + (3.80 + 3.18i)13-s + (3.11 − 5.39i)17-s + (−0.514 − 0.891i)19-s + (−0.602 + 3.41i)23-s + (−3.82 + 3.20i)25-s + (3.04 − 2.55i)29-s + (−0.740 + 4.20i)31-s + (−0.0481 − 0.0834i)35-s + (4.19 − 7.26i)37-s + (−2.15 − 1.80i)41-s + (−6.32 − 2.30i)43-s + (1.35 + 7.70i)47-s + ⋯
L(s)  = 1  + (−0.0426 + 0.0155i)5-s + (0.0623 + 0.353i)7-s + (1.14 + 0.417i)11-s + (1.05 + 0.884i)13-s + (0.755 − 1.30i)17-s + (−0.118 − 0.204i)19-s + (−0.125 + 0.712i)23-s + (−0.764 + 0.641i)25-s + (0.566 − 0.475i)29-s + (−0.133 + 0.754i)31-s + (−0.00814 − 0.0141i)35-s + (0.689 − 1.19i)37-s + (−0.335 − 0.281i)41-s + (−0.964 − 0.351i)43-s + (0.198 + 1.12i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 - 0.321i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 - 0.321i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.946 - 0.321i$
Analytic conductor: \(2.58715\)
Root analytic conductor: \(1.60846\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1/2),\ 0.946 - 0.321i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39774 + 0.230665i\)
\(L(\frac12)\) \(\approx\) \(1.39774 + 0.230665i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0952 - 0.0346i)T + (3.83 - 3.21i)T^{2} \)
7 \( 1 + (-0.165 - 0.935i)T + (-6.57 + 2.39i)T^{2} \)
11 \( 1 + (-3.80 - 1.38i)T + (8.42 + 7.07i)T^{2} \)
13 \( 1 + (-3.80 - 3.18i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-3.11 + 5.39i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.514 + 0.891i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.602 - 3.41i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (-3.04 + 2.55i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (0.740 - 4.20i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (-4.19 + 7.26i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.15 + 1.80i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (6.32 + 2.30i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-1.35 - 7.70i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + 12.9T + 53T^{2} \)
59 \( 1 + (13.3 - 4.87i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (0.469 + 2.66i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (2.53 + 2.12i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-7.26 + 12.5i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.82 + 10.0i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.981 - 0.823i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (4.02 - 3.37i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (-4.80 - 8.32i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.70 + 1.34i)T + (74.3 + 62.3i)T^{2} \)
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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

−11.72943164414607541284886249074, −10.91703304824542876626251826574, −9.467321640909154478314736860404, −9.154820741311495998438805320418, −7.81857434356897971345787730044, −6.80845405515517879419279303485, −5.82966274262068120809583745979, −4.52717610351088057237123946850, −3.36469825708802786426953066970, −1.60913658208964353157380493623, 1.30972759723857914254377215171, 3.33278001833116163454457673086, 4.26943602635983740189959894250, 5.86667636001125180398277082911, 6.51089682649884623350133585440, 8.018219257593038263236033580108, 8.523189866688102943625596796653, 9.852014691573615374736784677426, 10.61392982789048090769385616373, 11.53752299985574998278366514002

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