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Properties

Label 2-39-39.8-c3-0-1
Degree $2$
Conductor $39$
Sign $-0.521 - 0.852i$
Analytic cond. $2.30107$
Root an. cond. $1.51692$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.19·3-s + 8i·4-s + (−25.5 + 25.5i)7-s + 27·9-s − 41.5i·12-s + (31.1 − 35i)13-s − 64·16-s + (49.9 + 49.9i)19-s + (132. − 132. i)21-s + 125i·25-s − 140.·27-s + (−204. − 204. i)28-s + (231. + 231. i)31-s + 216i·36-s + (−163. + 163. i)37-s + ⋯
L(s)  = 1  − 1.00·3-s + i·4-s + (−1.38 + 1.38i)7-s + 9-s − 0.999i·12-s + (0.665 − 0.746i)13-s − 16-s + (0.603 + 0.603i)19-s + (1.38 − 1.38i)21-s + i·25-s − 1.00·27-s + (−1.38 − 1.38i)28-s + (1.34 + 1.34i)31-s + i·36-s + (−0.725 + 0.725i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.521 - 0.852i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.521 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $-0.521 - 0.852i$
Analytic conductor: \(2.30107\)
Root analytic conductor: \(1.51692\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :3/2),\ -0.521 - 0.852i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.336299 + 0.600030i\)
\(L(\frac12)\) \(\approx\) \(0.336299 + 0.600030i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 5.19T \)
13 \( 1 + (-31.1 + 35i)T \)
good2 \( 1 - 8iT^{2} \)
5 \( 1 - 125iT^{2} \)
7 \( 1 + (25.5 - 25.5i)T - 343iT^{2} \)
11 \( 1 + 1.33e3iT^{2} \)
17 \( 1 + 4.91e3T^{2} \)
19 \( 1 + (-49.9 - 49.9i)T + 6.85e3iT^{2} \)
23 \( 1 + 1.21e4T^{2} \)
29 \( 1 - 2.43e4T^{2} \)
31 \( 1 + (-231. - 231. i)T + 2.97e4iT^{2} \)
37 \( 1 + (163. - 163. i)T - 5.06e4iT^{2} \)
41 \( 1 - 6.89e4iT^{2} \)
43 \( 1 + 218. iT - 7.95e4T^{2} \)
47 \( 1 + 1.03e5iT^{2} \)
53 \( 1 - 1.48e5T^{2} \)
59 \( 1 + 2.05e5iT^{2} \)
61 \( 1 - 935.T + 2.26e5T^{2} \)
67 \( 1 + (-112. - 112. i)T + 3.00e5iT^{2} \)
71 \( 1 - 3.57e5iT^{2} \)
73 \( 1 + (407. - 407. i)T - 3.89e5iT^{2} \)
79 \( 1 + 1.09e3T + 4.93e5T^{2} \)
83 \( 1 - 5.71e5iT^{2} \)
89 \( 1 + 7.04e5iT^{2} \)
97 \( 1 + (-20.8 - 20.8i)T + 9.12e5iT^{2} \)
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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

−16.01574432766853867475195726766, −15.61816686332591610876954141705, −13.25731912265037693548921827382, −12.45585653964159330141840146826, −11.67247590116403927743192449948, −10.01009997448948708431825629532, −8.601769481558970937853468140015, −6.82179053380510227777450328860, −5.55138954005615592418663929417, −3.28808317311945961300263030970, 0.66419974701740016513030352077, 4.29787692687587276082604923756, 6.12679362412166326822274238597, 6.96658573105494420842171104876, 9.606147852636555739952188230441, 10.38085660668039823711397929570, 11.47602965085814111631916940223, 13.11492955502056318331339053989, 13.95877025949219821621648341835, 15.74919646684040088193110665381

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