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Properties

Label 2-39-13.10-c3-0-6
Degree $2$
Conductor $39$
Sign $0.318 + 0.947i$
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  + (2.83 − 1.63i)2-s + (−1.5 − 2.59i)3-s + (1.37 − 2.38i)4-s − 17.5i·5-s + (−8.51 − 4.91i)6-s + (23.1 + 13.3i)7-s + 17.2i·8-s + (−4.5 + 7.79i)9-s + (−28.7 − 49.8i)10-s + (−18.5 + 10.7i)11-s − 8.25·12-s + (−8.67 + 46.0i)13-s + 87.5·14-s + (−45.5 + 26.3i)15-s + (39.2 + 67.9i)16-s + (41.9 − 72.7i)17-s + ⋯
L(s)  = 1  + (1.00 − 0.579i)2-s + (−0.288 − 0.499i)3-s + (0.171 − 0.297i)4-s − 1.56i·5-s + (−0.579 − 0.334i)6-s + (1.24 + 0.720i)7-s + 0.760i·8-s + (−0.166 + 0.288i)9-s + (−0.909 − 1.57i)10-s + (−0.508 + 0.293i)11-s − 0.198·12-s + (−0.185 + 0.982i)13-s + 1.67·14-s + (−0.784 + 0.452i)15-s + (0.612 + 1.06i)16-s + (0.598 − 1.03i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.318 + 0.947i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.318 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $0.318 + 0.947i$
Analytic conductor: \(2.30107\)
Root analytic conductor: \(1.51692\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :3/2),\ 0.318 + 0.947i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.55913 - 1.12037i\)
\(L(\frac12)\) \(\approx\) \(1.55913 - 1.12037i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 + 2.59i)T \)
13 \( 1 + (8.67 - 46.0i)T \)
good2 \( 1 + (-2.83 + 1.63i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + 17.5iT - 125T^{2} \)
7 \( 1 + (-23.1 - 13.3i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (18.5 - 10.7i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (-41.9 + 72.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (66.7 + 38.5i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-71.0 - 123. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (67.1 + 116. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 122. iT - 2.97e4T^{2} \)
37 \( 1 + (192. - 111. i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (171. - 99.1i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-77.3 + 133. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 78.7iT - 1.03e5T^{2} \)
53 \( 1 + 477.T + 1.48e5T^{2} \)
59 \( 1 + (-37.1 - 21.4i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (248. - 430. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-419. + 242. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-331. - 191. i)T + (1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 193. iT - 3.89e5T^{2} \)
79 \( 1 - 1.04e3T + 4.93e5T^{2} \)
83 \( 1 + 861. iT - 5.71e5T^{2} \)
89 \( 1 + (-838. + 483. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-512. - 295. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−15.27538770030180111610259017742, −13.89795276036421937095699113241, −12.99720681537841113868494075752, −11.98321847996374727587660780745, −11.45657624881961387003566068254, −9.060742622333519877142375014609, −7.912621509092410559269024062726, −5.34060731012546637470117807706, −4.68588259576242002982920848933, −1.90578490940552884622991132825, 3.55048663572512894362889438130, 5.12391530416364200893398983388, 6.51043441218519952892767477024, 7.86417299323478364910932246348, 10.52529685770209444364509273938, 10.71143895704887832170523366389, 12.65268933095334689780240203737, 14.23025255082637238012423026370, 14.63429400744360897198610593483, 15.45274775797283156556236343328

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