Properties

Label 2-39-39.2-c3-0-4
Degree $2$
Conductor $39$
Sign $0.246 - 0.969i$
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  + (−0.532 + 1.98i)2-s + (4.60 + 2.40i)3-s + (3.26 + 1.88i)4-s + (−4.41 − 4.41i)5-s + (−7.22 + 7.86i)6-s + (3.85 − 1.03i)7-s + (−17.1 + 17.1i)8-s + (15.4 + 22.1i)9-s + (11.1 − 6.41i)10-s + (−5.64 − 1.51i)11-s + (10.5 + 16.5i)12-s + (−11.9 − 45.3i)13-s + 8.21i·14-s + (−9.72 − 30.9i)15-s + (−9.78 − 16.9i)16-s + (53.6 − 92.8i)17-s + ⋯
L(s)  = 1  + (−0.188 + 0.702i)2-s + (0.886 + 0.462i)3-s + (0.408 + 0.235i)4-s + (−0.394 − 0.394i)5-s + (−0.491 + 0.535i)6-s + (0.208 − 0.0558i)7-s + (−0.756 + 0.756i)8-s + (0.571 + 0.820i)9-s + (0.351 − 0.203i)10-s + (−0.154 − 0.0414i)11-s + (0.252 + 0.398i)12-s + (−0.254 − 0.967i)13-s + 0.156i·14-s + (−0.167 − 0.532i)15-s + (−0.152 − 0.264i)16-s + (0.764 − 1.32i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.246 - 0.969i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.246 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $0.246 - 0.969i$
Analytic conductor: \(2.30107\)
Root analytic conductor: \(1.51692\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :3/2),\ 0.246 - 0.969i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.23570 + 0.960434i\)
\(L(\frac12)\) \(\approx\) \(1.23570 + 0.960434i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.60 - 2.40i)T \)
13 \( 1 + (11.9 + 45.3i)T \)
good2 \( 1 + (0.532 - 1.98i)T + (-6.92 - 4i)T^{2} \)
5 \( 1 + (4.41 + 4.41i)T + 125iT^{2} \)
7 \( 1 + (-3.85 + 1.03i)T + (297. - 171.5i)T^{2} \)
11 \( 1 + (5.64 + 1.51i)T + (1.15e3 + 665.5i)T^{2} \)
17 \( 1 + (-53.6 + 92.8i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (15.7 + 58.6i)T + (-5.94e3 + 3.42e3i)T^{2} \)
23 \( 1 + (-56.5 - 97.9i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (118. - 68.6i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (10.7 - 10.7i)T - 2.97e4iT^{2} \)
37 \( 1 + (55.7 - 208. i)T + (-4.38e4 - 2.53e4i)T^{2} \)
41 \( 1 + (-83.3 + 311. i)T + (-5.96e4 - 3.44e4i)T^{2} \)
43 \( 1 + (-62.0 - 35.7i)T + (3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (258. - 258. i)T - 1.03e5iT^{2} \)
53 \( 1 - 216. iT - 1.48e5T^{2} \)
59 \( 1 + (166. + 621. i)T + (-1.77e5 + 1.02e5i)T^{2} \)
61 \( 1 + (371. - 644. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (335. + 90.0i)T + (2.60e5 + 1.50e5i)T^{2} \)
71 \( 1 + (-347. + 93.1i)T + (3.09e5 - 1.78e5i)T^{2} \)
73 \( 1 + (-761. - 761. i)T + 3.89e5iT^{2} \)
79 \( 1 - 670.T + 4.93e5T^{2} \)
83 \( 1 + (-1.03e3 - 1.03e3i)T + 5.71e5iT^{2} \)
89 \( 1 + (841. + 225. i)T + (6.10e5 + 3.52e5i)T^{2} \)
97 \( 1 + (-9.67 - 36.1i)T + (-7.90e5 + 4.56e5i)T^{2} \)
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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

−15.81773811228907703289116647876, −15.16562525989445652910002063910, −13.97080370309919851205613712299, −12.50937536813526410924493860260, −11.06959231907076739347056690898, −9.427390016677087895629390214694, −8.165525331352375669114291385106, −7.32758611865875054069349475915, −5.13273049283696957079050924863, −3.01709653053684032408196853165, 1.87021835888657170802200412786, 3.57864867616396155168027321626, 6.47571278867397614199518928649, 7.83546817859355007804599651527, 9.347029374203751187689695833420, 10.62330999485097930646570797719, 11.88978666878584841698318442513, 12.89830518005789621971033113296, 14.56820808666822218431500321097, 15.06577062978749654178436645932

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