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Properties

Label 2-39-39.2-c1-0-0
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: $\chi_{39} (2, \cdot )$
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

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

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