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Properties

Label 2-18e2-9.7-c1-0-1
Degree $2$
Conductor $324$
Sign $0.173 - 0.984i$
Analytic cond. $2.58715$
Root an. cond. $1.60846$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.5 + 2.59i)5-s + (−1 + 1.73i)7-s + (−3 + 5.19i)11-s + (−2.5 − 4.33i)13-s + 3·17-s + 2·19-s + (3 + 5.19i)23-s + (−2 + 3.46i)25-s + (1.5 − 2.59i)29-s + (2 + 3.46i)31-s − 6·35-s + 5·37-s + (−3 − 5.19i)41-s + (5 − 8.66i)43-s + (1.50 + 2.59i)49-s + ⋯
L(s)  = 1  + (0.670 + 1.16i)5-s + (−0.377 + 0.654i)7-s + (−0.904 + 1.56i)11-s + (−0.693 − 1.20i)13-s + 0.727·17-s + 0.458·19-s + (0.625 + 1.08i)23-s + (−0.400 + 0.692i)25-s + (0.278 − 0.482i)29-s + (0.359 + 0.622i)31-s − 1.01·35-s + 0.821·37-s + (−0.468 − 0.811i)41-s + (0.762 − 1.32i)43-s + (0.214 + 0.371i)49-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.173 - 0.984i$
Analytic conductor: \(2.58715\)
Root analytic conductor: \(1.60846\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1/2),\ 0.173 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.983282 + 0.825071i\)
\(L(\frac12)\) \(\approx\) \(0.983282 + 0.825071i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1 - 1.73i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (3 - 5.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.5 + 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 5T + 37T^{2} \)
41 \( 1 + (3 + 5.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5 + 8.66i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + T + 73T^{2} \)
79 \( 1 + (-5 + 8.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 3T + 89T^{2} \)
97 \( 1 + (-5 + 8.66i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.96297238117988667640298414736, −10.53374388284959934229764616553, −10.11664526213416518522555434750, −9.323598198158614980138775093847, −7.72041112268156637662184917747, −7.14427200571498360767566607545, −5.86595632836788444094026650577, −5.06591642150914633137430871306, −3.13604724997641892669080831300, −2.31321220030172474812161336887, 0.957563683851807472073539154453, 2.84814439494514772339290462718, 4.42296694578054834532579966763, 5.40080429617027325624458101214, 6.40285866834164375620609452158, 7.70411241119005063996128701604, 8.698233963198732900725570156511, 9.494133237303159615432348337833, 10.37864646771819563033577462290, 11.41470032453232554750095195641

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