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Properties

Label 2-82-41.27-c4-0-5
Degree $2$
Conductor $82$
Sign $0.680 - 0.732i$
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 + (2.35 − 0.974i)3-s + 8i·4-s + (−5.07 + 5.07i)5-s + (−6.65 − 2.75i)6-s + (13.7 − 5.69i)7-s + (16 − 16i)8-s + (−52.6 + 52.6i)9-s + 20.2·10-s + (53.0 + 128. i)11-s + (7.79 + 18.8i)12-s + (−78.5 + 32.5i)13-s + (−38.8 − 16.1i)14-s + (−6.98 + 16.8i)15-s − 64·16-s + (187. + 77.7i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (0.261 − 0.108i)3-s + 0.5i·4-s + (−0.202 + 0.202i)5-s + (−0.184 − 0.0765i)6-s + (0.280 − 0.116i)7-s + (0.250 − 0.250i)8-s + (−0.650 + 0.650i)9-s + 0.202·10-s + (0.438 + 1.05i)11-s + (0.0541 + 0.130i)12-s + (−0.465 + 0.192i)13-s + (−0.198 − 0.0821i)14-s + (−0.0310 + 0.0749i)15-s − 0.250·16-s + (0.649 + 0.268i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.680 - 0.732i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.680 - 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(82\)    =    \(2 \cdot 41\)
Sign: $0.680 - 0.732i$
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: \((2,\ 82,\ (\ :2),\ 0.680 - 0.732i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.07865 + 0.470410i\)
\(L(\frac12)\) \(\approx\) \(1.07865 + 0.470410i\)
\(L(3)\) 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 \( 1 + (-1.51e3 + 730. i)T \)
good3 \( 1 + (-2.35 + 0.974i)T + (57.2 - 57.2i)T^{2} \)
5 \( 1 + (5.07 - 5.07i)T - 625iT^{2} \)
7 \( 1 + (-13.7 + 5.69i)T + (1.69e3 - 1.69e3i)T^{2} \)
11 \( 1 + (-53.0 - 128. i)T + (-1.03e4 + 1.03e4i)T^{2} \)
13 \( 1 + (78.5 - 32.5i)T + (2.01e4 - 2.01e4i)T^{2} \)
17 \( 1 + (-187. - 77.7i)T + (5.90e4 + 5.90e4i)T^{2} \)
19 \( 1 + (-542. - 224. i)T + (9.21e4 + 9.21e4i)T^{2} \)
23 \( 1 - 173. iT - 2.79e5T^{2} \)
29 \( 1 + (-745. + 308. i)T + (5.00e5 - 5.00e5i)T^{2} \)
31 \( 1 - 515. iT - 9.23e5T^{2} \)
37 \( 1 + 2.58e3T + 1.87e6T^{2} \)
43 \( 1 + (949. + 949. i)T + 3.41e6iT^{2} \)
47 \( 1 + (1.90e3 + 789. i)T + (3.45e6 + 3.45e6i)T^{2} \)
53 \( 1 + (448. + 1.08e3i)T + (-5.57e6 + 5.57e6i)T^{2} \)
59 \( 1 + 3.21e3T + 1.21e7T^{2} \)
61 \( 1 + (-1.48e3 - 1.48e3i)T + 1.38e7iT^{2} \)
67 \( 1 + (-1.62e3 - 674. i)T + (1.42e7 + 1.42e7i)T^{2} \)
71 \( 1 + (-4.66e3 + 1.93e3i)T + (1.79e7 - 1.79e7i)T^{2} \)
73 \( 1 + (-1.84e3 - 1.84e3i)T + 2.83e7iT^{2} \)
79 \( 1 + (782. + 1.88e3i)T + (-2.75e7 + 2.75e7i)T^{2} \)
83 \( 1 - 2.50e3T + 4.74e7T^{2} \)
89 \( 1 + (-6.72e3 + 2.78e3i)T + (4.43e7 - 4.43e7i)T^{2} \)
97 \( 1 + (-1.15e3 + 2.78e3i)T + (-6.25e7 - 6.25e7i)T^{2} \)
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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

−13.81196798858697785022932414785, −12.35207469550541778132759311027, −11.59790403988357847722120445371, −10.36676121760830478978884036557, −9.368595158510675031223332208500, −8.035563186176324245610030647261, −7.16605076169191955497782107590, −5.12913237067639462973763508804, −3.35539709134853310565187349206, −1.70115964401999361610039916047, 0.70592335398933533383553775152, 3.17072429080308475301394798795, 5.13620760522457102661524143598, 6.44369375143678551470254011429, 7.915748236076828458482836708176, 8.818813769198509019018135523246, 9.814804457266512124701312512328, 11.28783657616525048561949069003, 12.17046847799080556668918076824, 13.90547599398752666081923559156

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