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Properties

Label 2-39-13.12-c3-0-4
Degree $2$
Conductor $39$
Sign $0.847 + 0.531i$
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  − 1.52i·2-s + 3·3-s + 5.67·4-s + 9.65i·5-s − 4.57i·6-s − 22.3i·7-s − 20.8i·8-s + 9·9-s + 14.7·10-s + 50.3i·11-s + 17.0·12-s + (−39.7 − 24.8i)13-s − 34.0·14-s + 28.9i·15-s + 13.6·16-s − 86.1·17-s + ⋯
L(s)  = 1  − 0.538i·2-s + 0.577·3-s + 0.709·4-s + 0.863i·5-s − 0.310i·6-s − 1.20i·7-s − 0.921i·8-s + 0.333·9-s + 0.465·10-s + 1.37i·11-s + 0.409·12-s + (−0.847 − 0.531i)13-s − 0.650·14-s + 0.498i·15-s + 0.213·16-s − 1.22·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.847 + 0.531i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.847 + 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $0.847 + 0.531i$
Analytic conductor: \(2.30107\)
Root analytic conductor: \(1.51692\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :3/2),\ 0.847 + 0.531i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.61189 - 0.463421i\)
\(L(\frac12)\) \(\approx\) \(1.61189 - 0.463421i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3T \)
13 \( 1 + (39.7 + 24.8i)T \)
good2 \( 1 + 1.52iT - 8T^{2} \)
5 \( 1 - 9.65iT - 125T^{2} \)
7 \( 1 + 22.3iT - 343T^{2} \)
11 \( 1 - 50.3iT - 1.33e3T^{2} \)
17 \( 1 + 86.1T + 4.91e3T^{2} \)
19 \( 1 - 116. iT - 6.85e3T^{2} \)
23 \( 1 + 72T + 1.21e4T^{2} \)
29 \( 1 - 14.1T + 2.43e4T^{2} \)
31 \( 1 + 196. iT - 2.97e4T^{2} \)
37 \( 1 - 154. iT - 5.06e4T^{2} \)
41 \( 1 + 265. iT - 6.89e4T^{2} \)
43 \( 1 + 211.T + 7.95e4T^{2} \)
47 \( 1 + 67.5iT - 1.03e5T^{2} \)
53 \( 1 - 686.T + 1.48e5T^{2} \)
59 \( 1 - 91.9iT - 2.05e5T^{2} \)
61 \( 1 - 329.T + 2.26e5T^{2} \)
67 \( 1 - 768. iT - 3.00e5T^{2} \)
71 \( 1 - 264. iT - 3.57e5T^{2} \)
73 \( 1 + 771. iT - 3.89e5T^{2} \)
79 \( 1 - 1.22e3T + 4.93e5T^{2} \)
83 \( 1 + 514. iT - 5.71e5T^{2} \)
89 \( 1 + 527. iT - 7.04e5T^{2} \)
97 \( 1 + 74.2iT - 9.12e5T^{2} \)
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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

−15.33654014751977186370184650853, −14.60059115800415823426312566289, −13.21491690717872269182295711793, −11.97843303298816643056632639017, −10.47200883765903349932721561952, −9.999201359348123166662267019749, −7.60769323165356513521241440575, −6.81759044521424932484784760918, −3.96217904350876123059012326389, −2.24565420657345228525944196038, 2.49482206629706922676557397874, 5.14729400854818857779589962431, 6.66188265347249156107358484717, 8.393218968189281304341396093993, 9.073746586965703148575828022714, 11.17801342567541892636590597802, 12.27284260297104871519878326131, 13.64459952880044262298616978040, 14.94577283398419799346550390836, 15.83823089385002916442091542516

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