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Properties

Label 2-18e2-36.7-c2-0-16
Degree $2$
Conductor $324$
Sign $-0.0962 - 0.995i$
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  + (1.85 + 0.756i)2-s + (2.85 + 2.80i)4-s + (3.70 + 6.41i)5-s + (−8.20 − 4.73i)7-s + (3.16 + 7.34i)8-s + (2.00 + 14.6i)10-s + (5.86 + 3.38i)11-s + (7.20 + 12.4i)13-s + (−11.6 − 14.9i)14-s + (0.291 + 15.9i)16-s − 17.8·17-s − 5.19i·19-s + (−7.40 + 28.6i)20-s + (8.29 + 10.7i)22-s + (21.5 − 12.4i)23-s + ⋯
L(s)  = 1  + (0.925 + 0.378i)2-s + (0.713 + 0.700i)4-s + (0.740 + 1.28i)5-s + (−1.17 − 0.677i)7-s + (0.395 + 0.918i)8-s + (0.200 + 1.46i)10-s + (0.533 + 0.307i)11-s + (0.554 + 0.960i)13-s + (−0.829 − 1.07i)14-s + (0.0182 + 0.999i)16-s − 1.05·17-s − 0.273i·19-s + (−0.370 + 1.43i)20-s + (0.376 + 0.486i)22-s + (0.938 − 0.542i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0962 - 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0962 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.0962 - 0.995i$
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: \((2,\ 324,\ (\ :1),\ -0.0962 - 0.995i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.95629 + 2.15455i\)
\(L(\frac12)\) \(\approx\) \(1.95629 + 2.15455i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.85 - 0.756i)T \)
3 \( 1 \)
good5 \( 1 + (-3.70 - 6.41i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (8.20 + 4.73i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-5.86 - 3.38i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-7.20 - 12.4i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 17.8T + 289T^{2} \)
19 \( 1 + 5.19iT - 361T^{2} \)
23 \( 1 + (-21.5 + 12.4i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-14.8 + 25.6i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (14.8 - 8.56i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 - 6.41T + 1.36e3T^{2} \)
41 \( 1 + (-4.32 - 7.48i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-43.4 - 25.0i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (45.0 + 26.0i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 82.6T + 2.80e3T^{2} \)
59 \( 1 + (-72.5 + 41.8i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-4.20 + 7.28i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-25.1 + 14.5i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 113. iT - 5.04e3T^{2} \)
73 \( 1 - 2.16T + 5.32e3T^{2} \)
79 \( 1 + (129. + 74.7i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (11.7 + 6.77i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 17.8T + 7.92e3T^{2} \)
97 \( 1 + (-69.1 + 119. i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−11.58625887624597379690623814700, −10.86338541743082715072557157273, −9.949000947140600317796749716928, −8.873638139013917463250138685923, −7.14833272683813818238092025789, −6.67574146125583556625045981279, −6.12675771397205842158662121697, −4.43540617126064433534943300960, −3.39825728997135922002209290237, −2.28780765618496013841311827566, 1.08900733925357888618651767168, 2.65457154949337291670395594915, 3.91217863337799101736891298349, 5.29929644063736603376270726805, 5.85449928066804653172177829482, 6.85998253988144370196924502748, 8.684484263708894837050610407078, 9.297882332176712344174099853790, 10.25133765605435456624555410445, 11.35514878662612937888543471530

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