Properties

Label 2-18e2-108.31-c2-0-11
Degree $2$
Conductor $324$
Sign $-0.714 - 0.699i$
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  + (−1.63 + 1.14i)2-s + (1.37 − 3.75i)4-s + (−1.09 + 6.21i)5-s + (5.65 + 6.74i)7-s + (2.05 + 7.73i)8-s + (−5.32 − 11.4i)10-s + (3.98 − 0.702i)11-s + (11.3 − 4.11i)13-s + (−17.0 − 4.56i)14-s + (−12.2 − 10.3i)16-s + (7.63 − 13.2i)17-s + (−20.6 + 11.9i)19-s + (21.8 + 12.6i)20-s + (−5.72 + 5.71i)22-s + (−11.4 + 13.6i)23-s + ⋯
L(s)  = 1  + (−0.819 + 0.573i)2-s + (0.343 − 0.939i)4-s + (−0.218 + 1.24i)5-s + (0.808 + 0.963i)7-s + (0.256 + 0.966i)8-s + (−0.532 − 1.14i)10-s + (0.362 − 0.0638i)11-s + (0.870 − 0.316i)13-s + (−1.21 − 0.326i)14-s + (−0.764 − 0.644i)16-s + (0.449 − 0.777i)17-s + (−1.08 + 0.626i)19-s + (1.09 + 0.631i)20-s + (−0.260 + 0.259i)22-s + (−0.499 + 0.595i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.714 - 0.699i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.714 - 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.714 - 0.699i$
Analytic conductor: \(8.82836\)
Root analytic conductor: \(2.97125\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1),\ -0.714 - 0.699i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.401410 + 0.984015i\)
\(L(\frac12)\) \(\approx\) \(0.401410 + 0.984015i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.63 - 1.14i)T \)
3 \( 1 \)
good5 \( 1 + (1.09 - 6.21i)T + (-23.4 - 8.55i)T^{2} \)
7 \( 1 + (-5.65 - 6.74i)T + (-8.50 + 48.2i)T^{2} \)
11 \( 1 + (-3.98 + 0.702i)T + (113. - 41.3i)T^{2} \)
13 \( 1 + (-11.3 + 4.11i)T + (129. - 108. i)T^{2} \)
17 \( 1 + (-7.63 + 13.2i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (20.6 - 11.9i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (11.4 - 13.6i)T + (-91.8 - 520. i)T^{2} \)
29 \( 1 + (-40.6 - 14.8i)T + (644. + 540. i)T^{2} \)
31 \( 1 + (31.8 - 38.0i)T + (-166. - 946. i)T^{2} \)
37 \( 1 + (14.4 - 24.9i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (40.7 - 14.8i)T + (1.28e3 - 1.08e3i)T^{2} \)
43 \( 1 + (-19.2 + 3.39i)T + (1.73e3 - 632. i)T^{2} \)
47 \( 1 + (-2.31 - 2.75i)T + (-383. + 2.17e3i)T^{2} \)
53 \( 1 + 81.1T + 2.80e3T^{2} \)
59 \( 1 + (48.7 + 8.58i)T + (3.27e3 + 1.19e3i)T^{2} \)
61 \( 1 + (-79.8 + 67.0i)T + (646. - 3.66e3i)T^{2} \)
67 \( 1 + (23.6 + 64.8i)T + (-3.43e3 + 2.88e3i)T^{2} \)
71 \( 1 + (-48.4 - 27.9i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-12.9 - 22.3i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (28.9 - 79.6i)T + (-4.78e3 - 4.01e3i)T^{2} \)
83 \( 1 + (21.4 - 58.8i)T + (-5.27e3 - 4.42e3i)T^{2} \)
89 \( 1 + (-11.6 - 20.1i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (16.4 + 93.0i)T + (-8.84e3 + 3.21e3i)T^{2} \)
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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

−11.37191848949249723146442075251, −10.80709665702375122075303037390, −9.887571329538588727237005560865, −8.677176203764794753483425966042, −8.096139731478360721339530013241, −6.94376255647268606063719985993, −6.16036997405369194260711196404, −5.07754075063494459492306552244, −3.18964303871423680828929877990, −1.72377787958668790436657812263, 0.69668269744801102934192337059, 1.80751318706560036154780577846, 3.91164156528373264189470702618, 4.54152325253996380595117957262, 6.32374090427133412141907606672, 7.61374197992080882901959573766, 8.433923730651276186585180138519, 8.964164857596102994610555040276, 10.23194774201421517096855852561, 10.95830448233162739675431027097

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