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Properties

Label 2-18e2-36.23-c1-0-1
Degree $2$
Conductor $324$
Sign $-0.819 - 0.573i$
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  + (−1.22 + 0.707i)2-s + (0.999 − 1.73i)4-s + (−2.56 − 1.48i)5-s + 2.82i·8-s + 4.19·10-s + (−3.59 + 6.23i)13-s + (−2.00 − 3.46i)16-s + 7.20i·17-s + (−5.13 + 2.96i)20-s + (1.90 + 3.29i)25-s − 10.1i·26-s + (−1.10 + 0.637i)29-s + (4.89 + 2.82i)32-s + (−5.09 − 8.83i)34-s − 11.3·37-s + ⋯
L(s)  = 1  + (−0.866 + 0.499i)2-s + (0.499 − 0.866i)4-s + (−1.14 − 0.663i)5-s + 0.999i·8-s + 1.32·10-s + (−0.997 + 1.72i)13-s + (−0.500 − 0.866i)16-s + 1.74i·17-s + (−1.14 + 0.663i)20-s + (0.380 + 0.658i)25-s − 1.99i·26-s + (−0.205 + 0.118i)29-s + (0.866 + 0.499i)32-s + (−0.874 − 1.51i)34-s − 1.87·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.819 - 0.573i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.819 - 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.819 - 0.573i$
Analytic conductor: \(2.58715\)
Root analytic conductor: \(1.60846\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1/2),\ -0.819 - 0.573i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0943903 + 0.299367i\)
\(L(\frac12)\) \(\approx\) \(0.0943903 + 0.299367i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 - 0.707i)T \)
3 \( 1 \)
good5 \( 1 + (2.56 + 1.48i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.59 - 6.23i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 7.20iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.10 - 0.637i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 11.3T + 37T^{2} \)
41 \( 1 + (1.22 + 0.707i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 7.07iT - 53T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.69 - 4.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 13.1T + 73T^{2} \)
79 \( 1 + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 5.51iT - 89T^{2} \)
97 \( 1 + (4 + 6.92i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.87784346912573915469101337494, −11.02794745984545256673256064881, −9.962669349565640928633185253532, −8.941756871963281535464781375589, −8.304962762780316911092849567443, −7.35421838885826749375428065237, −6.47351042775630099289319224833, −5.04221340443101416748257459680, −3.97847739302993635595370724472, −1.79709734849856550387382991267, 0.28733921152702162984702211301, 2.69895940520614535449122510669, 3.52620164722905384156990913473, 5.10781939722747765319429414615, 6.93614170242363476464358704900, 7.53360398768460653883832584425, 8.287612183007804846947375565663, 9.529123240141018634165645921963, 10.36837354888567429797741354190, 11.17031694769029671409519752641

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