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Properties

Label 2-39-13.2-c2-0-4
Degree $2$
Conductor $39$
Sign $0.987 - 0.155i$
Analytic cond. $1.06267$
Root an. cond. $1.03086$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.64 + 0.708i)2-s + (−0.866 − 1.5i)3-s + (3.02 + 1.74i)4-s + (−2.04 + 2.04i)5-s + (−1.22 − 4.58i)6-s + (−1.58 + 0.425i)7-s + (−0.976 − 0.976i)8-s + (−1.5 + 2.59i)9-s + (−6.84 + 3.95i)10-s + (−1.92 + 7.18i)11-s − 6.05i·12-s + (4.35 − 12.2i)13-s − 4.49·14-s + (4.83 + 1.29i)15-s + (−8.88 − 15.3i)16-s + (28.3 + 16.3i)17-s + ⋯
L(s)  = 1  + (1.32 + 0.354i)2-s + (−0.288 − 0.5i)3-s + (0.756 + 0.436i)4-s + (−0.408 + 0.408i)5-s + (−0.204 − 0.763i)6-s + (−0.226 + 0.0607i)7-s + (−0.122 − 0.122i)8-s + (−0.166 + 0.288i)9-s + (−0.684 + 0.395i)10-s + (−0.175 + 0.653i)11-s − 0.504i·12-s + (0.335 − 0.942i)13-s − 0.321·14-s + (0.322 + 0.0863i)15-s + (−0.555 − 0.961i)16-s + (1.66 + 0.964i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.155i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.987 - 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $0.987 - 0.155i$
Analytic conductor: \(1.06267\)
Root analytic conductor: \(1.03086\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :1),\ 0.987 - 0.155i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.60979 + 0.125702i\)
\(L(\frac12)\) \(\approx\) \(1.60979 + 0.125702i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 + 1.5i)T \)
13 \( 1 + (-4.35 + 12.2i)T \)
good2 \( 1 + (-2.64 - 0.708i)T + (3.46 + 2i)T^{2} \)
5 \( 1 + (2.04 - 2.04i)T - 25iT^{2} \)
7 \( 1 + (1.58 - 0.425i)T + (42.4 - 24.5i)T^{2} \)
11 \( 1 + (1.92 - 7.18i)T + (-104. - 60.5i)T^{2} \)
17 \( 1 + (-28.3 - 16.3i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (4.57 + 17.0i)T + (-312. + 180.5i)T^{2} \)
23 \( 1 + (16.1 - 9.33i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-9.45 - 16.3i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-32.0 + 32.0i)T - 961iT^{2} \)
37 \( 1 + (7.59 - 28.3i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (59.9 + 16.0i)T + (1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (20.9 + 12.0i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (26.1 + 26.1i)T + 2.20e3iT^{2} \)
53 \( 1 - 39.6T + 2.80e3T^{2} \)
59 \( 1 + (23.7 - 6.37i)T + (3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (-32.0 + 55.5i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (53.9 + 14.4i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + (-5.14 - 19.1i)T + (-4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (-34.2 - 34.2i)T + 5.32e3iT^{2} \)
79 \( 1 - 123.T + 6.24e3T^{2} \)
83 \( 1 + (19.4 - 19.4i)T - 6.88e3iT^{2} \)
89 \( 1 + (11.0 - 41.2i)T + (-6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (-38.2 - 142. i)T + (-8.14e3 + 4.70e3i)T^{2} \)
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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

−15.53131802604080853366747177767, −14.91577880129130858237635319547, −13.59399836492853129784753963524, −12.68418972641261448076672910756, −11.75831850168629236571284426421, −10.13894183851796570426234367815, −7.87586104117498385648493381603, −6.54227386635914256183863393072, −5.26712493231216613304175394503, −3.40520169230906670985266890756, 3.45602403801542101822473736027, 4.78446001133377040796705133991, 6.15368899542812986699708098809, 8.410856334198367661170008154798, 10.13162836787433984899732031802, 11.69067236173553318540718301216, 12.23562065548897161558303830840, 13.73414594645662760179227724271, 14.49435953178205267110700607455, 15.97467851762503176459749819758

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