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Properties

Label 2-18e2-12.11-c3-0-30
Degree $2$
Conductor $324$
Sign $0.356 + 0.934i$
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  + (−0.513 − 2.78i)2-s + (−7.47 + 2.85i)4-s + 2.40i·5-s − 2.65i·7-s + (11.7 + 19.3i)8-s + (6.70 − 1.23i)10-s − 48.2·11-s + 40.7·13-s + (−7.39 + 1.36i)14-s + (47.6 − 42.6i)16-s − 36.3i·17-s + 125. i·19-s + (−6.87 − 18.0i)20-s + (24.7 + 134. i)22-s + 194.·23-s + ⋯
L(s)  = 1  + (−0.181 − 0.983i)2-s + (−0.934 + 0.356i)4-s + 0.215i·5-s − 0.143i·7-s + (0.520 + 0.853i)8-s + (0.211 − 0.0391i)10-s − 1.32·11-s + 0.868·13-s + (−0.141 + 0.0260i)14-s + (0.745 − 0.666i)16-s − 0.518i·17-s + 1.51i·19-s + (−0.0769 − 0.201i)20-s + (0.240 + 1.30i)22-s + 1.75·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.356 + 0.934i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.356 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.356 + 0.934i$
Analytic conductor: \(19.1166\)
Root analytic conductor: \(4.37225\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :3/2),\ 0.356 + 0.934i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.417528262\)
\(L(\frac12)\) \(\approx\) \(1.417528262\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.513 + 2.78i)T \)
3 \( 1 \)
good5 \( 1 - 2.40iT - 125T^{2} \)
7 \( 1 + 2.65iT - 343T^{2} \)
11 \( 1 + 48.2T + 1.33e3T^{2} \)
13 \( 1 - 40.7T + 2.19e3T^{2} \)
17 \( 1 + 36.3iT - 4.91e3T^{2} \)
19 \( 1 - 125. iT - 6.85e3T^{2} \)
23 \( 1 - 194.T + 1.21e4T^{2} \)
29 \( 1 + 177. iT - 2.43e4T^{2} \)
31 \( 1 + 175. iT - 2.97e4T^{2} \)
37 \( 1 - 199.T + 5.06e4T^{2} \)
41 \( 1 + 233. iT - 6.89e4T^{2} \)
43 \( 1 - 291. iT - 7.95e4T^{2} \)
47 \( 1 - 61.8T + 1.03e5T^{2} \)
53 \( 1 + 352. iT - 1.48e5T^{2} \)
59 \( 1 + 141.T + 2.05e5T^{2} \)
61 \( 1 - 15.4T + 2.26e5T^{2} \)
67 \( 1 - 152. iT - 3.00e5T^{2} \)
71 \( 1 - 28.0T + 3.57e5T^{2} \)
73 \( 1 - 124.T + 3.89e5T^{2} \)
79 \( 1 + 748. iT - 4.93e5T^{2} \)
83 \( 1 - 348.T + 5.71e5T^{2} \)
89 \( 1 - 416. iT - 7.04e5T^{2} \)
97 \( 1 - 1.50e3T + 9.12e5T^{2} \)
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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

−10.91213792454571673111808028744, −10.30208994666562550989263303740, −9.316830806554047326108680792898, −8.275362824036951608899538724080, −7.47643065340401467027621988527, −5.87609620087099496163419021275, −4.75490537938584829857652699199, −3.48916884853883747365340393884, −2.41723479743398967351321010792, −0.803278166258825505838416134904, 0.916845216566830930541371009862, 3.04986830917482063311740709734, 4.70509819709750591052769786755, 5.39176653824804200844240937141, 6.62797219507168997473733880006, 7.45936751390814461356135879384, 8.650494107166783982616286284139, 9.028754824861882832543820568362, 10.46180407944122430299772627309, 11.02508298112002283966564065626

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