Properties

Degree $2$
Conductor $108$
Sign $-0.0796 - 0.996i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.63 + 1.03i)2-s + (5.16 − 0.555i)3-s + (5.86 − 5.43i)4-s + (6.86 + 18.8i)5-s + (−13.0 + 6.79i)6-s + (−18.0 + 3.17i)7-s + (−9.82 + 20.3i)8-s + (26.3 − 5.73i)9-s + (−37.5 − 42.5i)10-s + (−8.64 − 3.14i)11-s + (27.2 − 31.3i)12-s + (23.2 + 19.4i)13-s + (44.1 − 26.9i)14-s + (45.9 + 93.6i)15-s + (4.82 − 63.8i)16-s + (26.8 + 15.4i)17-s + ⋯
L(s)  = 1  + (−0.930 + 0.365i)2-s + (0.994 − 0.106i)3-s + (0.733 − 0.679i)4-s + (0.614 + 1.68i)5-s + (−0.886 + 0.462i)6-s + (−0.972 + 0.171i)7-s + (−0.434 + 0.900i)8-s + (0.977 − 0.212i)9-s + (−1.18 − 1.34i)10-s + (−0.237 − 0.0862i)11-s + (0.656 − 0.754i)12-s + (0.495 + 0.415i)13-s + (0.842 − 0.514i)14-s + (0.790 + 1.61i)15-s + (0.0754 − 0.997i)16-s + (0.382 + 0.221i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0796 - 0.996i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0796 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $-0.0796 - 0.996i$
Motivic weight: \(3\)
Character: $\chi_{108} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 108,\ (\ :3/2),\ -0.0796 - 0.996i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.950910 + 1.02996i\)
\(L(\frac12)\) \(\approx\) \(0.950910 + 1.02996i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.63 - 1.03i)T \)
3 \( 1 + (-5.16 + 0.555i)T \)
good5 \( 1 + (-6.86 - 18.8i)T + (-95.7 + 80.3i)T^{2} \)
7 \( 1 + (18.0 - 3.17i)T + (322. - 117. i)T^{2} \)
11 \( 1 + (8.64 + 3.14i)T + (1.01e3 + 855. i)T^{2} \)
13 \( 1 + (-23.2 - 19.4i)T + (381. + 2.16e3i)T^{2} \)
17 \( 1 + (-26.8 - 15.4i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (113. - 65.3i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-9.01 + 51.1i)T + (-1.14e4 - 4.16e3i)T^{2} \)
29 \( 1 + (-166. - 198. i)T + (-4.23e3 + 2.40e4i)T^{2} \)
31 \( 1 + (-176. - 31.0i)T + (2.79e4 + 1.01e4i)T^{2} \)
37 \( 1 + (-134. + 232. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (36.5 - 43.5i)T + (-1.19e4 - 6.78e4i)T^{2} \)
43 \( 1 + (-120. + 331. i)T + (-6.09e4 - 5.11e4i)T^{2} \)
47 \( 1 + (-41.3 - 234. i)T + (-9.75e4 + 3.55e4i)T^{2} \)
53 \( 1 + 424. iT - 1.48e5T^{2} \)
59 \( 1 + (79.6 - 28.9i)T + (1.57e5 - 1.32e5i)T^{2} \)
61 \( 1 + (143. + 813. i)T + (-2.13e5 + 7.76e4i)T^{2} \)
67 \( 1 + (-328. + 390. i)T + (-5.22e4 - 2.96e5i)T^{2} \)
71 \( 1 + (-37.9 + 65.6i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (340. + 589. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-756. - 901. i)T + (-8.56e4 + 4.85e5i)T^{2} \)
83 \( 1 + (-163. + 136. i)T + (9.92e4 - 5.63e5i)T^{2} \)
89 \( 1 + (-719. + 415. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-571. - 207. i)T + (6.99e5 + 5.86e5i)T^{2} \)
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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

−13.95675525718761661050627865667, −12.54963340881943047649128686463, −10.75544099871270988555185366365, −10.20775938952040043619316803299, −9.233227996792134505317024259761, −8.060763644740017972508944529661, −6.74975134590521760418342525094, −6.28319184113022120021792837483, −3.27961239148486963515328691511, −2.16513653678960378689794739410, 0.963316385047051414469324795133, 2.64084966500236073458448170820, 4.34647293528992680715443823691, 6.33284310712940736140326315849, 7.984873674253504117407975436996, 8.740091454865314790171967934702, 9.583724209563777019369547906758, 10.26805529675745415450136517584, 12.08115447269212362145814973306, 13.16136778985452533599499312636

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