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Properties

Label 2-39-13.9-c3-0-0
Degree $2$
Conductor $39$
Sign $-0.999 - 0.0256i$
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  + (−1.5 + 2.59i)2-s + (−1.5 + 2.59i)3-s + (−0.5 − 0.866i)4-s − 9·5-s + (−4.5 − 7.79i)6-s + (−1 − 1.73i)7-s − 21·8-s + (−4.5 − 7.79i)9-s + (13.5 − 23.3i)10-s + (−15 + 25.9i)11-s + 3.00·12-s + (32.5 + 33.7i)13-s + 6·14-s + (13.5 − 23.3i)15-s + (35.5 − 61.4i)16-s + (55.5 + 96.1i)17-s + ⋯
L(s)  = 1  + (−0.530 + 0.918i)2-s + (−0.288 + 0.499i)3-s + (−0.0625 − 0.108i)4-s − 0.804·5-s + (−0.306 − 0.530i)6-s + (−0.0539 − 0.0935i)7-s − 0.928·8-s + (−0.166 − 0.288i)9-s + (0.426 − 0.739i)10-s + (−0.411 + 0.712i)11-s + 0.0721·12-s + (0.693 + 0.720i)13-s + 0.114·14-s + (0.232 − 0.402i)15-s + (0.554 − 0.960i)16-s + (0.791 + 1.37i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0256i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $-0.999 - 0.0256i$
Analytic conductor: \(2.30107\)
Root analytic conductor: \(1.51692\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :3/2),\ -0.999 - 0.0256i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.00810706 + 0.632160i\)
\(L(\frac12)\) \(\approx\) \(0.00810706 + 0.632160i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 - 2.59i)T \)
13 \( 1 + (-32.5 - 33.7i)T \)
good2 \( 1 + (1.5 - 2.59i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + 9T + 125T^{2} \)
7 \( 1 + (1 + 1.73i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (15 - 25.9i)T + (-665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (-55.5 - 96.1i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-23 - 39.8i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-52.5 + 90.9i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 100T + 2.97e4T^{2} \)
37 \( 1 + (8.5 - 14.7i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-115.5 + 200. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-257 - 445. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 162T + 1.03e5T^{2} \)
53 \( 1 - 639T + 1.48e5T^{2} \)
59 \( 1 + (300 + 519. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (116.5 + 201. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (463 - 801. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-465 - 805. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + 253T + 3.89e5T^{2} \)
79 \( 1 + 1.32e3T + 4.93e5T^{2} \)
83 \( 1 - 810T + 5.71e5T^{2} \)
89 \( 1 + (249 - 431. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (679 + 1.17e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−16.24969277513667795997208849911, −15.50821788485238729494204240858, −14.54885741006108173524894542380, −12.58044740512525358456738198264, −11.48386637473184268745797676856, −9.978461626752906803190522287028, −8.515736016715523211199606996911, −7.44163828027592214396493997798, −5.95406759271731495737716299802, −3.89582735149094349531468270883, 0.66173629771368750311017618197, 3.07740421174496972295053312568, 5.69451087536718866451308193320, 7.55338517681390666667414049695, 8.949347206956979154885561145505, 10.53413358628060379033607761988, 11.45846709897553172883995877490, 12.31903367622654020636877500208, 13.72168642900760615102413011191, 15.35502232383205681484682292362

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