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Properties

Label 2-18e2-9.7-c3-0-4
Degree $2$
Conductor $324$
Sign $0.984 + 0.173i$
Analytic cond. $19.1166$
Root an. cond. $4.37225$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.33 − 9.24i)5-s + (−11.7 + 20.3i)7-s + (2.45 − 4.25i)11-s + (−0.752 − 1.30i)13-s + 99.2·17-s + 28.5·19-s + (76.3 + 132. i)23-s + (5.48 − 9.50i)25-s + (120. − 208. i)29-s + (−128. − 222. i)31-s + 250.·35-s + 359.·37-s + (−54.2 − 93.9i)41-s + (−205. + 356. i)43-s + (155. − 270i)47-s + ⋯
L(s)  = 1  + (−0.477 − 0.827i)5-s + (−0.634 + 1.09i)7-s + (0.0672 − 0.116i)11-s + (−0.0160 − 0.0277i)13-s + 1.41·17-s + 0.344·19-s + (0.692 + 1.19i)23-s + (0.0438 − 0.0760i)25-s + (0.769 − 1.33i)29-s + (−0.744 − 1.28i)31-s + 1.21·35-s + 1.59·37-s + (−0.206 − 0.357i)41-s + (−0.729 + 1.26i)43-s + (0.483 − 0.837i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.984 + 0.173i$
Analytic conductor: \(19.1166\)
Root analytic conductor: \(4.37225\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :3/2),\ 0.984 + 0.173i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.577746697\)
\(L(\frac12)\) \(\approx\) \(1.577746697\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (5.33 + 9.24i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 + (11.7 - 20.3i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-2.45 + 4.25i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (0.752 + 1.30i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 99.2T + 4.91e3T^{2} \)
19 \( 1 - 28.5T + 6.85e3T^{2} \)
23 \( 1 + (-76.3 - 132. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-120. + 208. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (128. + 222. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 359.T + 5.06e4T^{2} \)
41 \( 1 + (54.2 + 93.9i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (205. - 356. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-155. + 270i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 702.T + 1.48e5T^{2} \)
59 \( 1 + (-239. - 414. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (249. - 431. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-166. - 287. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 884.T + 3.57e5T^{2} \)
73 \( 1 - 305T + 3.89e5T^{2} \)
79 \( 1 + (-351. + 609. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-210. + 364. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 1.60e3T + 7.04e5T^{2} \)
97 \( 1 + (198. - 342. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−11.52161605334582303660886037169, −9.958798316485431649820691916523, −9.314481707016922091446182783565, −8.360942827048946271229511080214, −7.50266157570999974944912576572, −6.02588663543302948690708235682, −5.30677860883952659415010006493, −3.94193175218541128251614568552, −2.67407058673458532971182270584, −0.856095681072410385770658158573, 0.901487896417828839303380766430, 3.00422289412848975349956309640, 3.77789027186043707073241908151, 5.17744425308133110231167355028, 6.72177270737092926874367594887, 7.13523668767303832425452465555, 8.223325100571977948058275064199, 9.530555200576437397378130331734, 10.45699735134523928246381699045, 10.94277620898385649319210901553

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