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Properties

Label 2-39-39.29-c4-0-16
Degree $2$
Conductor $39$
Sign $0.230 + 0.973i$
Analytic cond. $4.03142$
Root an. cond. $2.00784$
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  + (5.96 − 3.44i)2-s + (4.35 − 7.87i)3-s + (15.7 − 27.2i)4-s + 27.0i·5-s + (−1.14 − 61.9i)6-s + (−39.3 + 68.0i)7-s − 106. i·8-s + (−43.0 − 68.6i)9-s + (93.2 + 161. i)10-s + (−3.21 + 1.85i)11-s + (−145. − 242. i)12-s + (150. − 76.4i)13-s + 541. i·14-s + (213. + 117. i)15-s + (−114. − 198. i)16-s + (−325. − 187. i)17-s + ⋯
L(s)  = 1  + (1.49 − 0.860i)2-s + (0.483 − 0.875i)3-s + (0.982 − 1.70i)4-s + 1.08i·5-s + (−0.0317 − 1.72i)6-s + (−0.802 + 1.38i)7-s − 1.66i·8-s + (−0.531 − 0.846i)9-s + (0.932 + 1.61i)10-s + (−0.0266 + 0.0153i)11-s + (−1.01 − 1.68i)12-s + (0.891 − 0.452i)13-s + 2.76i·14-s + (0.948 + 0.524i)15-s + (−0.446 − 0.774i)16-s + (−1.12 − 0.649i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 + 0.973i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.230 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $0.230 + 0.973i$
Analytic conductor: \(4.03142\)
Root analytic conductor: \(2.00784\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :2),\ 0.230 + 0.973i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.52548 - 1.99707i\)
\(L(\frac12)\) \(\approx\) \(2.52548 - 1.99707i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.35 + 7.87i)T \)
13 \( 1 + (-150. + 76.4i)T \)
good2 \( 1 + (-5.96 + 3.44i)T + (8 - 13.8i)T^{2} \)
5 \( 1 - 27.0iT - 625T^{2} \)
7 \( 1 + (39.3 - 68.0i)T + (-1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (3.21 - 1.85i)T + (7.32e3 - 1.26e4i)T^{2} \)
17 \( 1 + (325. + 187. i)T + (4.17e4 + 7.23e4i)T^{2} \)
19 \( 1 + (-116. + 201. i)T + (-6.51e4 - 1.12e5i)T^{2} \)
23 \( 1 + (-48.8 + 28.2i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (1.06e3 - 613. i)T + (3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + 41.3T + 9.23e5T^{2} \)
37 \( 1 + (-776. - 1.34e3i)T + (-9.37e5 + 1.62e6i)T^{2} \)
41 \( 1 + (-619. + 357. i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-1.30e3 + 2.25e3i)T + (-1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + 2.25e3iT - 4.87e6T^{2} \)
53 \( 1 - 2.15e3iT - 7.89e6T^{2} \)
59 \( 1 + (1.99e3 + 1.15e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (1.69e3 - 2.94e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-1.08e3 - 1.88e3i)T + (-1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (3.20e3 + 1.85e3i)T + (1.27e7 + 2.20e7i)T^{2} \)
73 \( 1 + 2.59e3T + 2.83e7T^{2} \)
79 \( 1 - 1.16e4T + 3.89e7T^{2} \)
83 \( 1 - 3.68e3iT - 4.74e7T^{2} \)
89 \( 1 + (1.82e3 - 1.05e3i)T + (3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 + (-415. + 720. i)T + (-4.42e7 - 7.66e7i)T^{2} \)
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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

−14.94522748904469015939689858689, −13.70576877667005859720008923950, −12.97419818427828615969140254147, −11.91105299926117621258679161892, −10.92284844583912819947307770945, −9.062404481613693603261099793461, −6.80186071586555542020208821122, −5.76736315832023288422966962281, −3.26616332029495772443107582164, −2.40324948498638914509577842867, 3.74262563328163175127928213165, 4.48414853894267824919478717751, 6.11889217028187850928590609322, 7.77879257262258912438282912603, 9.321799742679432886421834313006, 11.02648637643149031563722898857, 12.91932079024253106439543253529, 13.43616730973288669891168709705, 14.46814491833367784991566699497, 15.79573264315744433770859946401

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