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Properties

Label 2-39-39.32-c1-0-2
Degree $2$
Conductor $39$
Sign $0.993 + 0.112i$
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 + (−1.60 + 0.650i)3-s + (0.232 − 0.133i)4-s + (−1.06 − 1.06i)5-s + (−2.08 + 1.57i)6-s + (0.366 − 1.36i)7-s + (−1.84 + 1.84i)8-s + (2.15 − 2.08i)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 + (2.40 + 1.01i)15-s + (−2.23 + 3.86i)16-s + (−2.51 − 4.36i)17-s + ⋯
L(s)  = 1  + (1.02 − 0.275i)2-s + (−0.926 + 0.375i)3-s + (0.116 − 0.0669i)4-s + (−0.476 − 0.476i)5-s + (−0.849 + 0.641i)6-s + (0.138 − 0.516i)7-s + (−0.652 + 0.652i)8-s + (0.717 − 0.696i)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 + (0.620 + 0.262i)15-s + (−0.558 + 0.966i)16-s + (−0.611 − 1.05i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.112i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $0.993 + 0.112i$
Analytic conductor: \(0.311416\)
Root analytic conductor: \(0.558047\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :1/2),\ 0.993 + 0.112i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.870195 - 0.0491120i\)
\(L(\frac12)\) \(\approx\) \(0.870195 - 0.0491120i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.60 - 0.650i)T \)
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

−16.10002393394389896673356486142, −15.12679366765685008263782838910, −13.72778464604812677950385580670, −12.54492875281927302651405320281, −11.79422637659007618863192518385, −10.57328804431090635588794925255, −8.840826222319201104905626971520, −6.74412489325006960992443518691, −4.95599865204656729193849253605, −4.08471658587654049200097916742, 3.94754720668736229970447449288, 5.72918964101977330788315056569, 6.56108889588994508196361140845, 8.552475945350572964937770686574, 10.71727803952668282784591808159, 11.69998262647513641089449588237, 12.86835351478038347661834432237, 13.84444802202909022779840548534, 15.16967356146586055994965872926, 16.00310175815050686451821204837

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