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Properties

Label 2-18e2-36.31-c2-0-0
Degree $2$
Conductor $324$
Sign $-0.993 + 0.116i$
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  + (−1.91 − 0.573i)2-s + (3.34 + 2.19i)4-s + (−2.18 + 3.78i)5-s + (1.84 − 1.06i)7-s + (−5.14 − 6.12i)8-s + (6.36 − 6.00i)10-s + (−10.9 + 6.30i)11-s + (−4.63 + 8.02i)13-s + (−4.15 + 0.984i)14-s + (6.33 + 14.6i)16-s + 14.6·17-s − 34.5i·19-s + (−15.6 + 7.85i)20-s + (24.5 − 5.81i)22-s + (−35.3 − 20.3i)23-s + ⋯
L(s)  = 1  + (−0.957 − 0.286i)2-s + (0.835 + 0.549i)4-s + (−0.437 + 0.757i)5-s + (0.264 − 0.152i)7-s + (−0.642 − 0.766i)8-s + (0.636 − 0.600i)10-s + (−0.993 + 0.573i)11-s + (−0.356 + 0.617i)13-s + (−0.296 + 0.0703i)14-s + (0.396 + 0.918i)16-s + 0.861·17-s − 1.81i·19-s + (−0.781 + 0.392i)20-s + (1.11 − 0.264i)22-s + (−1.53 − 0.886i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.116i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.993 + 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.993 + 0.116i$
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: \((2,\ 324,\ (\ :1),\ -0.993 + 0.116i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.000994364 - 0.0170748i\)
\(L(\frac12)\) \(\approx\) \(0.000994364 - 0.0170748i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.91 + 0.573i)T \)
3 \( 1 \)
good5 \( 1 + (2.18 - 3.78i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (-1.84 + 1.06i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (10.9 - 6.30i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (4.63 - 8.02i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 14.6T + 289T^{2} \)
19 \( 1 + 34.5iT - 361T^{2} \)
23 \( 1 + (35.3 + 20.3i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (9.52 + 16.4i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (0.860 + 0.496i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + 66.4T + 1.36e3T^{2} \)
41 \( 1 + (12.9 - 22.4i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (36.4 - 21.0i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-30.1 + 17.3i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 12.2T + 2.80e3T^{2} \)
59 \( 1 + (49.1 + 28.3i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (36.8 + 63.8i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (82.4 + 47.5i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 75.5iT - 5.04e3T^{2} \)
73 \( 1 + 56.7T + 5.32e3T^{2} \)
79 \( 1 + (-64.5 + 37.2i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-56.7 + 32.7i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 150.T + 7.92e3T^{2} \)
97 \( 1 + (-56.4 - 97.7i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−11.67039099335590019737165723771, −10.78455240370048067707681711689, −10.15047952049403893560682548148, −9.164153966410456472685438570262, −7.978571116318691729721108721410, −7.35476514922645168819707101916, −6.45531461313527430619331660943, −4.75918954289174240881378131990, −3.24226506328250683745324420240, −2.10217957478837978210832754921, 0.01048425907924287681321207820, 1.66584185086688782797132152978, 3.41712546629884416637601065328, 5.24812765241880377255025155462, 5.87579596316802170646170991376, 7.56284303508558395949234498609, 8.041896044559294638691695050857, 8.801743666844785407492758616820, 10.11319939743872676915946989661, 10.51875804946675731868930095210

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