Properties

Label 2-18e2-36.31-c2-0-39
Degree $2$
Conductor $324$
Sign $0.939 + 0.342i$
Analytic cond. $8.82836$
Root an. cond. $2.97125$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 4·4-s + (1 − 1.73i)5-s + (6 − 3.46i)7-s + 8·8-s + (2 − 3.46i)10-s + (−6 + 3.46i)11-s + (−1 + 1.73i)13-s + (12 − 6.92i)14-s + 16·16-s + 10·17-s − 20.7i·19-s + (4 − 6.92i)20-s + (−12 + 6.92i)22-s + (−24 − 13.8i)23-s + ⋯
L(s)  = 1  + 2-s + 4-s + (0.200 − 0.346i)5-s + (0.857 − 0.494i)7-s + 8-s + (0.200 − 0.346i)10-s + (−0.545 + 0.314i)11-s + (−0.0769 + 0.133i)13-s + (0.857 − 0.494i)14-s + 16-s + 0.588·17-s − 1.09i·19-s + (0.200 − 0.346i)20-s + (−0.545 + 0.314i)22-s + (−1.04 − 0.602i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.939 + 0.342i$
Analytic conductor: \(8.82836\)
Root analytic conductor: \(2.97125\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1),\ 0.939 + 0.342i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.37377 - 0.594886i\)
\(L(\frac12)\) \(\approx\) \(3.37377 - 0.594886i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2T \)
3 \( 1 \)
good5 \( 1 + (-1 + 1.73i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (-6 + 3.46i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (6 - 3.46i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 10T + 289T^{2} \)
19 \( 1 + 20.7iT - 361T^{2} \)
23 \( 1 + (24 + 13.8i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-13 - 22.5i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-6 - 3.46i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 - 26T + 1.36e3T^{2} \)
41 \( 1 + (29 - 50.2i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-42 + 24.2i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (60 - 34.6i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 74T + 2.80e3T^{2} \)
59 \( 1 + (78 + 45.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (13 + 22.5i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-6 - 3.46i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + 46T + 5.32e3T^{2} \)
79 \( 1 + (102 - 58.8i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (42 - 24.2i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 82T + 7.92e3T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−11.38450157546239039523420068052, −10.72708856817599293653862684837, −9.672432328679105085508528927590, −8.221324849462244508245156427103, −7.42232202005322908874275671669, −6.31726624825752320991366781181, −5.04791950917796816308725988536, −4.50308972591366444918788554264, −2.96266961726695260431721378248, −1.49738287502713629169316420775, 1.83627215909546123013354652938, 3.06211886144995575029150057950, 4.40030990816663310216133463887, 5.53234246021590886110051399242, 6.20722678184229780528759230076, 7.64291755911515966455585913226, 8.251667108022740096739230768903, 9.930262170102490234114113184780, 10.67183922304515160595772983314, 11.70175612991605185008161303429

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