Properties

Label 2-39-39.5-c3-0-2
Degree $2$
Conductor $39$
Sign $0.978 + 0.208i$
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  + (−3.62 − 3.62i)2-s + (4.19 + 3.07i)3-s + 18.2i·4-s + (10.8 + 10.8i)5-s + (−4.05 − 26.3i)6-s + (−5.52 − 5.52i)7-s + (37.1 − 37.1i)8-s + (8.11 + 25.7i)9-s − 78.9i·10-s + (25.6 − 25.6i)11-s + (−56.1 + 76.5i)12-s + (6.01 + 46.4i)13-s + 40.0i·14-s + (12.1 + 79.1i)15-s − 123.·16-s − 56.1·17-s + ⋯
L(s)  = 1  + (−1.28 − 1.28i)2-s + (0.806 + 0.591i)3-s + 2.28i·4-s + (0.974 + 0.974i)5-s + (−0.275 − 1.79i)6-s + (−0.298 − 0.298i)7-s + (1.64 − 1.64i)8-s + (0.300 + 0.953i)9-s − 2.49i·10-s + (0.702 − 0.702i)11-s + (−1.35 + 1.84i)12-s + (0.128 + 0.991i)13-s + 0.764i·14-s + (0.209 + 1.36i)15-s − 1.92·16-s − 0.800·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.208i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.978 + 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $0.978 + 0.208i$
Analytic conductor: \(2.30107\)
Root analytic conductor: \(1.51692\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :3/2),\ 0.978 + 0.208i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.982587 - 0.103547i\)
\(L(\frac12)\) \(\approx\) \(0.982587 - 0.103547i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.19 - 3.07i)T \)
13 \( 1 + (-6.01 - 46.4i)T \)
good2 \( 1 + (3.62 + 3.62i)T + 8iT^{2} \)
5 \( 1 + (-10.8 - 10.8i)T + 125iT^{2} \)
7 \( 1 + (5.52 + 5.52i)T + 343iT^{2} \)
11 \( 1 + (-25.6 + 25.6i)T - 1.33e3iT^{2} \)
17 \( 1 + 56.1T + 4.91e3T^{2} \)
19 \( 1 + (-72.4 + 72.4i)T - 6.85e3iT^{2} \)
23 \( 1 + 8.33T + 1.21e4T^{2} \)
29 \( 1 + 125. iT - 2.43e4T^{2} \)
31 \( 1 + (89.8 - 89.8i)T - 2.97e4iT^{2} \)
37 \( 1 + (274. + 274. i)T + 5.06e4iT^{2} \)
41 \( 1 + (44.2 + 44.2i)T + 6.89e4iT^{2} \)
43 \( 1 - 100. iT - 7.95e4T^{2} \)
47 \( 1 + (-82.5 + 82.5i)T - 1.03e5iT^{2} \)
53 \( 1 - 31.7iT - 1.48e5T^{2} \)
59 \( 1 + (-150. + 150. i)T - 2.05e5iT^{2} \)
61 \( 1 - 383.T + 2.26e5T^{2} \)
67 \( 1 + (-227. + 227. i)T - 3.00e5iT^{2} \)
71 \( 1 + (-298. - 298. i)T + 3.57e5iT^{2} \)
73 \( 1 + (560. + 560. i)T + 3.89e5iT^{2} \)
79 \( 1 - 805.T + 4.93e5T^{2} \)
83 \( 1 + (828. + 828. i)T + 5.71e5iT^{2} \)
89 \( 1 + (373. - 373. i)T - 7.04e5iT^{2} \)
97 \( 1 + (-254. + 254. i)T - 9.12e5iT^{2} \)
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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

−16.04414624350346081066452450077, −14.21755729410932393412623974956, −13.40022171523488260208642911084, −11.45136131335261114619425289016, −10.55602639398446889189009705442, −9.543777624141278477849741206753, −8.828445818389145079762878157604, −7.00458701360794975172685312071, −3.57326074421142184281396547483, −2.17875027477075681236917098370, 1.43450353910668014596437465252, 5.60201490497414815892384077610, 6.90259487881726929124745821695, 8.280273112539328384576492702109, 9.191709054615361665758340211974, 9.943397485024478693078879844142, 12.53144407077574206095924254010, 13.75011438283667766450110856644, 14.88549647271682063349405311085, 15.91063375499830468259704920888

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