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Properties

Label 2-82-41.27-c4-0-7
Degree $2$
Conductor $82$
Sign $-0.214 + 0.976i$
Analytic cond. $8.47633$
Root an. cond. $2.91141$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 − 2i)2-s + (−4.49 + 1.85i)3-s + 8i·4-s + (−21.9 + 21.9i)5-s + (12.7 + 5.26i)6-s + (50.6 − 20.9i)7-s + (16 − 16i)8-s + (−40.5 + 40.5i)9-s + 87.6·10-s + (−63.9 − 154. i)11-s + (−14.8 − 35.9i)12-s + (216. − 89.6i)13-s + (−143. − 59.3i)14-s + (57.6 − 139. i)15-s − 64·16-s + (−214. − 88.8i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (−0.498 + 0.206i)3-s + 0.5i·4-s + (−0.876 + 0.876i)5-s + (0.352 + 0.146i)6-s + (1.03 − 0.428i)7-s + (0.250 − 0.250i)8-s + (−0.500 + 0.500i)9-s + 0.876·10-s + (−0.528 − 1.27i)11-s + (−0.103 − 0.249i)12-s + (1.28 − 0.530i)13-s + (−0.731 − 0.302i)14-s + (0.256 − 0.618i)15-s − 0.250·16-s + (−0.742 − 0.307i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.214 + 0.976i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.214 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(82\)    =    \(2 \cdot 41\)
Sign: $-0.214 + 0.976i$
Analytic conductor: \(8.47633\)
Root analytic conductor: \(2.91141\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{82} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 82,\ (\ :2),\ -0.214 + 0.976i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.408040 - 0.507256i\)
\(L(\frac12)\) \(\approx\) \(0.408040 - 0.507256i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2 + 2i)T \)
41 \( 1 + (933. - 1.39e3i)T \)
good3 \( 1 + (4.49 - 1.85i)T + (57.2 - 57.2i)T^{2} \)
5 \( 1 + (21.9 - 21.9i)T - 625iT^{2} \)
7 \( 1 + (-50.6 + 20.9i)T + (1.69e3 - 1.69e3i)T^{2} \)
11 \( 1 + (63.9 + 154. i)T + (-1.03e4 + 1.03e4i)T^{2} \)
13 \( 1 + (-216. + 89.6i)T + (2.01e4 - 2.01e4i)T^{2} \)
17 \( 1 + (214. + 88.8i)T + (5.90e4 + 5.90e4i)T^{2} \)
19 \( 1 + (124. + 51.5i)T + (9.21e4 + 9.21e4i)T^{2} \)
23 \( 1 + 744. iT - 2.79e5T^{2} \)
29 \( 1 + (-798. + 330. i)T + (5.00e5 - 5.00e5i)T^{2} \)
31 \( 1 + 1.26e3iT - 9.23e5T^{2} \)
37 \( 1 - 355.T + 1.87e6T^{2} \)
43 \( 1 + (-1.27e3 - 1.27e3i)T + 3.41e6iT^{2} \)
47 \( 1 + (-2.33e3 - 965. i)T + (3.45e6 + 3.45e6i)T^{2} \)
53 \( 1 + (1.62e3 + 3.91e3i)T + (-5.57e6 + 5.57e6i)T^{2} \)
59 \( 1 + 2.44e3T + 1.21e7T^{2} \)
61 \( 1 + (2.26e3 + 2.26e3i)T + 1.38e7iT^{2} \)
67 \( 1 + (-4.88e3 - 2.02e3i)T + (1.42e7 + 1.42e7i)T^{2} \)
71 \( 1 + (-170. + 70.8i)T + (1.79e7 - 1.79e7i)T^{2} \)
73 \( 1 + (7.23e3 + 7.23e3i)T + 2.83e7iT^{2} \)
79 \( 1 + (1.76e3 + 4.25e3i)T + (-2.75e7 + 2.75e7i)T^{2} \)
83 \( 1 + 1.06e4T + 4.74e7T^{2} \)
89 \( 1 + (1.51e3 - 628. i)T + (4.43e7 - 4.43e7i)T^{2} \)
97 \( 1 + (-49.9 + 120. i)T + (-6.25e7 - 6.25e7i)T^{2} \)
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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

−13.27413773320893088822432270868, −11.51701935323000063388085955253, −11.03817058409763282018708514387, −10.60923547628615270561714413052, −8.449200649359860593369335750748, −7.914812688840319353689508833737, −6.20869236563340411603231437607, −4.42349300156624110591800841407, −2.86743678161029056634780284590, −0.44265665391363622730092223850, 1.41653115589178672464514704125, 4.38472239241574393339151482528, 5.56333251052544199566276962279, 7.06533530690653683213595384625, 8.351984561954658432758174038580, 8.963894063574316024248725433162, 10.76678270891057564265238836719, 11.74495735405897722276676378909, 12.53177769006799844149369310472, 14.05646910484501464150116992161

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