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Properties

Label 2-39-39.11-c1-0-1
Degree $2$
Conductor $39$
Sign $0.351 + 0.936i$
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  + (−1.45 − 0.389i)2-s + (0.239 − 1.71i)3-s + (0.232 + 0.133i)4-s + (1.06 − 1.06i)5-s + (−1.01 + 2.40i)6-s + (0.366 + 1.36i)7-s + (1.84 + 1.84i)8-s + (−2.88 − 0.820i)9-s + (−1.96 + 1.13i)10-s + (−1.06 + 3.97i)11-s + (0.285 − 0.366i)12-s + (3.59 + 0.232i)13-s − 2.12i·14-s + (−1.57 − 2.08i)15-s + (−2.23 − 3.86i)16-s + (2.51 − 4.36i)17-s + ⋯
L(s)  = 1  + (−1.02 − 0.275i)2-s + (0.138 − 0.990i)3-s + (0.116 + 0.0669i)4-s + (0.476 − 0.476i)5-s + (−0.415 + 0.980i)6-s + (0.138 + 0.516i)7-s + (0.652 + 0.652i)8-s + (−0.961 − 0.273i)9-s + (−0.621 + 0.358i)10-s + (−0.321 + 1.19i)11-s + (0.0823 − 0.105i)12-s + (0.997 + 0.0643i)13-s − 0.569i·14-s + (−0.405 − 0.537i)15-s + (−0.558 − 0.966i)16-s + (0.611 − 1.05i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.351 + 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.351 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $0.351 + 0.936i$
Analytic conductor: \(0.311416\)
Root analytic conductor: \(0.558047\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :1/2),\ 0.351 + 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.421043 - 0.291630i\)
\(L(\frac12)\) \(\approx\) \(0.421043 - 0.291630i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.239 + 1.71i)T \)
13 \( 1 + (-3.59 - 0.232i)T \)
good2 \( 1 + (1.45 + 0.389i)T + (1.73 + i)T^{2} \)
5 \( 1 + (-1.06 + 1.06i)T - 5iT^{2} \)
7 \( 1 + (-0.366 - 1.36i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (1.06 - 3.97i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-2.51 + 4.36i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.73 - i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (6.20 - 3.58i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.46 + 2.46i)T + 31iT^{2} \)
37 \( 1 + (5.23 + 1.40i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-5.42 - 1.45i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-1.90 - 1.09i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.25 + 4.25i)T + 47iT^{2} \)
53 \( 1 + 0.779iT - 53T^{2} \)
59 \( 1 + (2.90 - 0.779i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.53 - 5.73i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (0.779 + 2.90i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (0.901 - 0.901i)T - 73iT^{2} \)
79 \( 1 - 2T + 79T^{2} \)
83 \( 1 + (-2.90 + 2.90i)T - 83iT^{2} \)
89 \( 1 + (2.41 - 9.01i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-1.63 + 0.437i)T + (84.0 - 48.5i)T^{2} \)
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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

−16.55743357445146294226353549882, −14.78615635204347166425771417192, −13.53082564970482701980803769968, −12.50570744403579383388635188000, −11.13681544742550342333398363301, −9.560487094476591237969049109584, −8.642274846801149240759873676364, −7.37041387682648272866778873398, −5.43884288613253686495927374509, −1.84816786409955800015425415144, 3.81856285014947763266534833046, 6.05142661502583991481500636523, 8.045450346075109607081572456416, 8.997619564720382669656620647576, 10.39841282818986344036276589992, 10.87756384267549753742568170851, 13.28951949705771400240830153780, 14.32023371589416464065032437022, 15.72051592257398388121122366497, 16.66238193703605810760542091106

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