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Properties

Label 2-39-39.35-c4-0-7
Degree $2$
Conductor $39$
Sign $-0.928 + 0.372i$
Analytic cond. $4.03142$
Root an. cond. $2.00784$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.70 − 3.29i)2-s + (2.61 + 8.61i)3-s + (13.6 + 23.7i)4-s − 26.9i·5-s + (13.4 − 57.7i)6-s + (−6.28 − 10.8i)7-s − 74.9i·8-s + (−67.3 + 45.0i)9-s + (−88.8 + 153. i)10-s + (−172. − 99.3i)11-s + (−168. + 179. i)12-s + (−58.7 − 158. i)13-s + 82.8i·14-s + (232. − 70.6i)15-s + (−27.7 + 48.1i)16-s + (−22.0 + 12.7i)17-s + ⋯
L(s)  = 1  + (−1.42 − 0.823i)2-s + (0.290 + 0.956i)3-s + (0.855 + 1.48i)4-s − 1.07i·5-s + (0.373 − 1.60i)6-s + (−0.128 − 0.222i)7-s − 1.17i·8-s + (−0.830 + 0.556i)9-s + (−0.888 + 1.53i)10-s + (−1.42 − 0.821i)11-s + (−1.16 + 1.24i)12-s + (−0.347 − 0.937i)13-s + 0.422i·14-s + (1.03 − 0.313i)15-s + (−0.108 + 0.188i)16-s + (−0.0761 + 0.0439i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.928 + 0.372i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.928 + 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $-0.928 + 0.372i$
Analytic conductor: \(4.03142\)
Root analytic conductor: \(2.00784\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :2),\ -0.928 + 0.372i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0662748 - 0.342958i\)
\(L(\frac12)\) \(\approx\) \(0.0662748 - 0.342958i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.61 - 8.61i)T \)
13 \( 1 + (58.7 + 158. i)T \)
good2 \( 1 + (5.70 + 3.29i)T + (8 + 13.8i)T^{2} \)
5 \( 1 + 26.9iT - 625T^{2} \)
7 \( 1 + (6.28 + 10.8i)T + (-1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (172. + 99.3i)T + (7.32e3 + 1.26e4i)T^{2} \)
17 \( 1 + (22.0 - 12.7i)T + (4.17e4 - 7.23e4i)T^{2} \)
19 \( 1 + (117. + 203. i)T + (-6.51e4 + 1.12e5i)T^{2} \)
23 \( 1 + (503. + 290. i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (-1.00e3 - 579. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 - 527.T + 9.23e5T^{2} \)
37 \( 1 + (1.25e3 - 2.17e3i)T + (-9.37e5 - 1.62e6i)T^{2} \)
41 \( 1 + (-1.33e3 - 770. i)T + (1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (1.07e3 + 1.85e3i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + 2.06e3iT - 4.87e6T^{2} \)
53 \( 1 - 2.74e3iT - 7.89e6T^{2} \)
59 \( 1 + (1.46e3 - 848. i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (1.84e3 + 3.20e3i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (1.78e3 - 3.08e3i)T + (-1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 + (927. - 535. i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 - 3.33e3T + 2.83e7T^{2} \)
79 \( 1 - 7.81e3T + 3.89e7T^{2} \)
83 \( 1 + 3.48e3iT - 4.74e7T^{2} \)
89 \( 1 + (13.8 + 8.01i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (6.35e3 + 1.10e4i)T + (-4.42e7 + 7.66e7i)T^{2} \)
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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

−15.52811613210561109617443530281, −13.55087048451685640458082394581, −12.19183992405002374871897960421, −10.67351395490176479058989282310, −10.09981029173746168232923014366, −8.674698299912736673609324249164, −8.141919308633856892713357686422, −5.07148291697620391830304765172, −2.86413618218794927002971057747, −0.33648483159444917443420104764, 2.25764218121256856821761608963, 6.19258422351861983258077356330, 7.23867441537986517121686854375, 8.053806508306590915504006112353, 9.560688210198766088205052235575, 10.70706268184926121075354363303, 12.34281967573212386595043420774, 14.00005502466061941248257013463, 15.04909990553405367472598898526, 16.06501063343905353915239418495

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