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Properties

Label 2-18e2-4.3-c4-0-88
Degree $2$
Conductor $324$
Sign $-0.529 + 0.848i$
Analytic cond. $33.4918$
Root an. cond. $5.78721$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.49 − 1.94i)2-s + (8.46 − 13.5i)4-s + 33.2·5-s − 46.1i·7-s + (3.25 − 63.9i)8-s + (116. − 64.4i)10-s − 73.4i·11-s − 303.·13-s + (−89.5 − 161. i)14-s + (−112. − 229. i)16-s + 182.·17-s − 314. i·19-s + (281. − 451. i)20-s + (−142. − 256. i)22-s + 335. i·23-s + ⋯
L(s)  = 1  + (0.874 − 0.485i)2-s + (0.529 − 0.848i)4-s + 1.32·5-s − 0.942i·7-s + (0.0508 − 0.998i)8-s + (1.16 − 0.644i)10-s − 0.607i·11-s − 1.79·13-s + (−0.457 − 0.823i)14-s + (−0.440 − 0.897i)16-s + 0.629·17-s − 0.870i·19-s + (0.703 − 1.12i)20-s + (−0.294 − 0.530i)22-s + 0.635i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.529 + 0.848i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.529 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.529 + 0.848i$
Analytic conductor: \(33.4918\)
Root analytic conductor: \(5.78721\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :2),\ -0.529 + 0.848i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(3.863815025\)
\(L(\frac12)\) \(\approx\) \(3.863815025\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.49 + 1.94i)T \)
3 \( 1 \)
good5 \( 1 - 33.2T + 625T^{2} \)
7 \( 1 + 46.1iT - 2.40e3T^{2} \)
11 \( 1 + 73.4iT - 1.46e4T^{2} \)
13 \( 1 + 303.T + 2.85e4T^{2} \)
17 \( 1 - 182.T + 8.35e4T^{2} \)
19 \( 1 + 314. iT - 1.30e5T^{2} \)
23 \( 1 - 335. iT - 2.79e5T^{2} \)
29 \( 1 - 714.T + 7.07e5T^{2} \)
31 \( 1 - 1.13e3iT - 9.23e5T^{2} \)
37 \( 1 - 1.00e3T + 1.87e6T^{2} \)
41 \( 1 + 1.11e3T + 2.82e6T^{2} \)
43 \( 1 + 2.51e3iT - 3.41e6T^{2} \)
47 \( 1 - 1.13e3iT - 4.87e6T^{2} \)
53 \( 1 - 1.05e3T + 7.89e6T^{2} \)
59 \( 1 + 1.01e3iT - 1.21e7T^{2} \)
61 \( 1 - 860.T + 1.38e7T^{2} \)
67 \( 1 - 645. iT - 2.01e7T^{2} \)
71 \( 1 + 9.56e3iT - 2.54e7T^{2} \)
73 \( 1 - 1.89e3T + 2.83e7T^{2} \)
79 \( 1 - 7.81e3iT - 3.89e7T^{2} \)
83 \( 1 + 8.14e3iT - 4.74e7T^{2} \)
89 \( 1 + 7.65e3T + 6.27e7T^{2} \)
97 \( 1 - 1.27e4T + 8.85e7T^{2} \)
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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

−10.50592292969912317157256612863, −10.06888996540655100406349367618, −9.193624795508470839083473368952, −7.43427272480862352140692340041, −6.58704808001673222891157565574, −5.46104799331019832916166746713, −4.71887826796473057431164697029, −3.23039428627976284146164579731, −2.14658471980172454161105943578, −0.811522064529038966671569302536, 2.04513621506473095341931403511, 2.73854867935985117045971914640, 4.54673586622203315890684425088, 5.45482946013206531822727058317, 6.14456955989441422803784643832, 7.24008960089685970666984884757, 8.306148963977737642029528839627, 9.583156424033289604261802448980, 10.13650048656225831994652152468, 11.70625377548457445360727527890

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