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Properties

Label 2-18e2-36.31-c2-0-35
Degree $2$
Conductor $324$
Sign $-0.766 + 0.642i$
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 + (4 − 6.92i)5-s + 7.99·8-s − 15.9·10-s + (5 − 8.66i)13-s + (−8 − 13.8i)16-s + 16·17-s + (15.9 + 27.7i)20-s + (−19.4 − 33.7i)25-s − 20·26-s + (−20 − 34.6i)29-s + (−15.9 + 27.7i)32-s + (−16 − 27.7i)34-s − 70·37-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.800 − 1.38i)5-s + 0.999·8-s − 1.59·10-s + (0.384 − 0.666i)13-s + (−0.5 − 0.866i)16-s + 0.941·17-s + (0.799 + 1.38i)20-s + (−0.779 − 1.35i)25-s − 0.769·26-s + (−0.689 − 1.19i)29-s + (−0.499 + 0.866i)32-s + (−0.470 − 0.815i)34-s − 1.89·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.766 + 0.642i$
Analytic conductor: \(8.82836\)
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: \((2,\ 324,\ (\ :1),\ -0.766 + 0.642i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.443925 - 1.21967i\)
\(L(\frac12)\) \(\approx\) \(0.443925 - 1.21967i\)
\(L(2)\) 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 \( 1 + (-4 + 6.92i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-5 + 8.66i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 16T + 289T^{2} \)
19 \( 1 - 361T^{2} \)
23 \( 1 + (264.5 + 458. i)T^{2} \)
29 \( 1 + (20 + 34.6i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (480.5 + 832. i)T^{2} \)
37 \( 1 + 70T + 1.36e3T^{2} \)
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 \( 1 + (-11 - 19.0i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 - 110T + 5.32e3T^{2} \)
79 \( 1 + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 160T + 7.92e3T^{2} \)
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

−10.91502636543359360975893234589, −9.995807900446350032472476256431, −9.271686117580921966436130774485, −8.491423163455395543887596065900, −7.60274418522020420947650050760, −5.82987594022744100465275374225, −4.90727822443623711545050827219, −3.60046653250209872314013993422, −1.98516170305361835744857367911, −0.76484877705502247109661328735, 1.74397715522564305210952759904, 3.41883690034154622110848871809, 5.14298408026605656423172404619, 6.19978390460462349539228098792, 6.84660815991702511611277764692, 7.77050590704299318437566350568, 9.007751727035655767094991427293, 9.854217296909285194459277376297, 10.57676282819587240169114653849, 11.38341182990168128216609776822

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