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Properties

Label 2-18e2-36.31-c2-0-31
Degree $2$
Conductor $324$
Sign $0.430 + 0.902i$
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  + (1.58 − 1.22i)2-s + (1.00 − 3.87i)4-s + (−3.70 + 6.41i)5-s + (8.20 − 4.73i)7-s + (−3.16 − 7.34i)8-s + (1.99 + 14.6i)10-s + (5.86 − 3.38i)11-s + (7.20 − 12.4i)13-s + (7.17 − 17.5i)14-s + (−13.9 − 7.74i)16-s + 17.8·17-s − 5.19i·19-s + (21.1 + 20.7i)20-s + (5.12 − 12.5i)22-s + (21.5 + 12.4i)23-s + ⋯
L(s)  = 1  + (0.790 − 0.612i)2-s + (0.250 − 0.968i)4-s + (−0.740 + 1.28i)5-s + (1.17 − 0.677i)7-s + (−0.395 − 0.918i)8-s + (0.199 + 1.46i)10-s + (0.533 − 0.307i)11-s + (0.554 − 0.960i)13-s + (0.512 − 1.25i)14-s + (−0.874 − 0.484i)16-s + 1.05·17-s − 0.273i·19-s + (1.05 + 1.03i)20-s + (0.232 − 0.569i)22-s + (0.938 + 0.542i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.430 + 0.902i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.430 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.430 + 0.902i$
Analytic conductor: \(8.82836\)
Root analytic conductor: \(2.97125\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1),\ 0.430 + 0.902i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.21982 - 1.39999i\)
\(L(\frac12)\) \(\approx\) \(2.21982 - 1.39999i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.58 + 1.22i)T \)
3 \( 1 \)
good5 \( 1 + (3.70 - 6.41i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (-8.20 + 4.73i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-5.86 + 3.38i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-7.20 + 12.4i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 17.8T + 289T^{2} \)
19 \( 1 + 5.19iT - 361T^{2} \)
23 \( 1 + (-21.5 - 12.4i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (14.8 + 25.6i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-14.8 - 8.56i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 - 6.41T + 1.36e3T^{2} \)
41 \( 1 + (4.32 - 7.48i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (43.4 - 25.0i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (45.0 - 26.0i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 82.6T + 2.80e3T^{2} \)
59 \( 1 + (-72.5 - 41.8i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-4.20 - 7.28i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (25.1 + 14.5i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 113. iT - 5.04e3T^{2} \)
73 \( 1 - 2.16T + 5.32e3T^{2} \)
79 \( 1 + (-129. + 74.7i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (11.7 - 6.77i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 17.8T + 7.92e3T^{2} \)
97 \( 1 + (-69.1 - 119. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−11.24426674290321990276107654058, −10.74420838881819556941308423417, −9.803805706049996669710556832844, −8.143662972121095341851612541359, −7.32117149966323428530178415793, −6.24863144863088929283214247853, −5.00476918663550773086925554435, −3.78590324604135125911009306341, −3.02093906181800227278008376668, −1.17588086147620793719981517604, 1.63371409745384988884036152580, 3.65846171821206918164953761012, 4.72680025937851731109375845308, 5.27219115384826750401304767299, 6.64600840443529245326595998078, 7.88719561237554158491033362241, 8.483352841411266780732936630129, 9.233434749172055827450770380834, 11.15692256691803372077444945285, 11.88498395609006169497986768286

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