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Properties

Label 2-39-13.9-c1-0-0
Degree $2$
Conductor $39$
Sign $-0.597 - 0.802i$
Analytic cond. $0.311416$
Root an. cond. $0.558047$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.28 + 2.21i)2-s + (−0.5 + 0.866i)3-s + (−2.28 − 3.95i)4-s + 0.561·5-s + (−1.28 − 2.21i)6-s + (1.78 + 3.08i)7-s + 6.56·8-s + (−0.499 − 0.866i)9-s + (−0.719 + 1.24i)10-s + (1 − 1.73i)11-s + 4.56·12-s + (0.5 − 3.57i)13-s − 9.12·14-s + (−0.280 + 0.486i)15-s + (−3.84 + 6.65i)16-s + (−1.28 − 2.21i)17-s + ⋯
L(s)  = 1  + (−0.905 + 1.56i)2-s + (−0.288 + 0.499i)3-s + (−1.14 − 1.97i)4-s + 0.251·5-s + (−0.522 − 0.905i)6-s + (0.673 + 1.16i)7-s + 2.31·8-s + (−0.166 − 0.288i)9-s + (−0.227 + 0.393i)10-s + (0.301 − 0.522i)11-s + 1.31·12-s + (0.138 − 0.990i)13-s − 2.43·14-s + (−0.0724 + 0.125i)15-s + (−0.960 + 1.66i)16-s + (−0.310 − 0.538i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.597 - 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.597 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $-0.597 - 0.802i$
Analytic conductor: \(0.311416\)
Root analytic conductor: \(0.558047\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :1/2),\ -0.597 - 0.802i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.221424 + 0.440843i\)
\(L(\frac12)\) \(\approx\) \(0.221424 + 0.440843i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-0.5 + 3.57i)T \)
good2 \( 1 + (1.28 - 2.21i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 0.561T + 5T^{2} \)
7 \( 1 + (-1.78 - 3.08i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.28 + 2.21i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.561 - 0.972i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1 - 1.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.84 + 4.92i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.56T + 31T^{2} \)
37 \( 1 + (1.71 - 2.97i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.28 - 2.21i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.219 + 0.379i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 8.24T + 47T^{2} \)
53 \( 1 - 11.6T + 53T^{2} \)
59 \( 1 + (-5.56 - 9.63i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.06 + 10.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.219 - 0.379i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (7 + 12.1i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 1.87T + 73T^{2} \)
79 \( 1 - 9.56T + 79T^{2} \)
83 \( 1 + 9.12T + 83T^{2} \)
89 \( 1 + (6.56 - 11.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.21 - 3.84i)T + (-48.5 + 84.0i)T^{2} \)
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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

−16.63626487814764417731471398613, −15.57813499304656079020648439903, −15.01285821919102920629488857213, −13.75745221868974112782545901432, −11.65894298933109523607952369996, −10.08321046175364581921012691368, −8.947228123916774784562183180588, −7.961425274891963000628013599450, −6.13080867782957238623680716028, −5.22682145542702382543323010512, 1.69954342443032167672892412906, 4.19046505752440223713785140131, 7.16476381016400549389812101067, 8.593978564601701248595039586842, 10.00983854422326925471949820373, 11.01306185846334421911876392666, 11.92839061625033255481999988837, 13.14467812921236505569820267523, 14.19216404935030640251005927216, 16.62083944234154749659699482497

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