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Properties

Label 2-18e2-36.7-c2-0-41
Degree $2$
Conductor $324$
Sign $-0.984 - 0.173i$
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  + (1 − 1.73i)2-s + (−1.99 − 3.46i)4-s + (−0.598 − 1.03i)5-s − 7.99·8-s − 2.39·10-s + (−12.8 − 22.3i)13-s + (−8 + 13.8i)16-s − 17.9·17-s + (−2.39 + 4.14i)20-s + (11.7 − 20.4i)25-s − 51.5·26-s + (−28.1 + 48.8i)29-s + (15.9 + 27.7i)32-s + (−17.9 + 31.1i)34-s + 55.7·37-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.119 − 0.207i)5-s − 0.999·8-s − 0.239·10-s + (−0.991 − 1.71i)13-s + (−0.5 + 0.866i)16-s − 1.05·17-s + (−0.119 + 0.207i)20-s + (0.471 − 0.816i)25-s − 1.98·26-s + (−0.971 + 1.68i)29-s + (0.499 + 0.866i)32-s + (−0.528 + 0.915i)34-s + 1.50·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.173i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.984 - 0.173i$
Analytic conductor: \(8.82836\)
Root analytic conductor: \(2.97125\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1),\ -0.984 - 0.173i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0999318 + 1.14222i\)
\(L(\frac12)\) \(\approx\) \(0.0999318 + 1.14222i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + 1.73i)T \)
3 \( 1 \)
good5 \( 1 + (0.598 + 1.03i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (60.5 + 104. i)T^{2} \)
13 \( 1 + (12.8 + 22.3i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 17.9T + 289T^{2} \)
19 \( 1 - 361T^{2} \)
23 \( 1 + (264.5 - 458. i)T^{2} \)
29 \( 1 + (28.1 - 48.8i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (480.5 - 832. i)T^{2} \)
37 \( 1 - 55.7T + 1.36e3T^{2} \)
41 \( 1 + (40 + 69.2i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 56T + 2.80e3T^{2} \)
59 \( 1 + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-46.4 + 80.4i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 - 28.1T + 5.32e3T^{2} \)
79 \( 1 + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 147.T + 7.92e3T^{2} \)
97 \( 1 + (-65 + 112. i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−10.86326840348364826010901629835, −10.25939116386552697484829908207, −9.220912193350662005508190788820, −8.246341770485677283225294874001, −6.92261068184372270744492572113, −5.55764884525515908690118602326, −4.78472198699973980187783933167, −3.46191523489280002114682951692, −2.27873611170642453589274816899, −0.43869045333165453002045447098, 2.44106473290052039714036845648, 4.04207731572283003254845633768, 4.82003449637990640836671278631, 6.19474849763016507358748106701, 6.97275689997351419817989493841, 7.83856658071600178700667449976, 9.040185566338129357073919967813, 9.685871381568808299637919126198, 11.34373042627687443484322245002, 11.82565536751431563898956977526

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