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Properties

Label 2-39-39.20-c1-0-2
Degree $2$
Conductor $39$
Sign $0.173 + 0.984i$
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  + (−0.619 − 2.31i)2-s + (1.64 + 0.529i)3-s + (−3.23 + 1.86i)4-s + (−1.69 + 1.69i)5-s + (0.202 − 4.14i)6-s + (−1.36 − 0.366i)7-s + (2.93 + 2.93i)8-s + (2.43 + 1.74i)9-s + (4.96 + 2.86i)10-s + (1.69 − 0.453i)11-s + (−6.31 + 1.36i)12-s + (−1.59 − 3.23i)13-s + 3.38i·14-s + (−3.68 + 1.89i)15-s + (1.23 − 2.13i)16-s + (−1.07 − 1.85i)17-s + ⋯
L(s)  = 1  + (−0.438 − 1.63i)2-s + (0.952 + 0.305i)3-s + (−1.61 + 0.933i)4-s + (−0.757 + 0.757i)5-s + (0.0826 − 1.69i)6-s + (−0.516 − 0.138i)7-s + (1.03 + 1.03i)8-s + (0.813 + 0.582i)9-s + (1.56 + 0.906i)10-s + (0.510 − 0.136i)11-s + (−1.82 + 0.394i)12-s + (−0.443 − 0.896i)13-s + 0.904i·14-s + (−0.952 + 0.489i)15-s + (0.308 − 0.533i)16-s + (−0.260 − 0.450i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $0.173 + 0.984i$
Analytic conductor: \(0.311416\)
Root analytic conductor: \(0.558047\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :1/2),\ 0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.527403 - 0.442641i\)
\(L(\frac12)\) \(\approx\) \(0.527403 - 0.442641i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.64 - 0.529i)T \)
13 \( 1 + (1.59 + 3.23i)T \)
good2 \( 1 + (0.619 + 2.31i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (1.69 - 1.69i)T - 5iT^{2} \)
7 \( 1 + (1.36 + 0.366i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-1.69 + 0.453i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (1.07 + 1.85i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.267 - i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.79 + 2.76i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.46 - 4.46i)T + 31iT^{2} \)
37 \( 1 + (1.76 + 6.59i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-0.166 - 0.619i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-7.09 + 4.09i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.77 - 6.77i)T + 47iT^{2} \)
53 \( 1 + 4.62iT - 53T^{2} \)
59 \( 1 + (1.23 - 4.62i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.46 - 2.26i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (4.62 + 1.23i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (6.09 - 6.09i)T - 73iT^{2} \)
79 \( 1 - 2T + 79T^{2} \)
83 \( 1 + (-1.23 + 1.23i)T - 83iT^{2} \)
89 \( 1 + (-9.70 + 2.60i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-3.36 + 12.5i)T + (-84.0 - 48.5i)T^{2} \)
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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

−15.88608965023622454100499163557, −14.70314119025926642192869132511, −13.41092794085449443966235085708, −12.26134608195860603413563426659, −10.97256281162903504664475116932, −10.04277744569358230029749645644, −8.927066501196046369859546013071, −7.47799291026492365417143583494, −3.93268514198483980193294624806, −2.83605023072791147539139764242, 4.37483855784114194924410955852, 6.50642671967879362419518081301, 7.65224176197566244321321150969, 8.733677732048123639644704330291, 9.507377539112253572026028854146, 12.14820953060878567606817791001, 13.45262995228327361181981503156, 14.61400383234057652572098996048, 15.47279570193605938860279934918, 16.35824710221340722564814347310

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