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Properties

Label 2-39-13.2-c2-0-3
Degree $2$
Conductor $39$
Sign $0.112 + 0.993i$
Analytic cond. $1.06267$
Root an. cond. $1.03086$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.942 − 0.252i)2-s + (−0.866 − 1.5i)3-s + (−2.63 − 1.52i)4-s + (6.89 − 6.89i)5-s + (0.437 + 1.63i)6-s + (−5.17 + 1.38i)7-s + (4.86 + 4.86i)8-s + (−1.5 + 2.59i)9-s + (−8.24 + 4.75i)10-s + (0.0813 − 0.303i)11-s + 5.27i·12-s + (12.0 + 4.88i)13-s + 5.22·14-s + (−16.3 − 4.37i)15-s + (2.74 + 4.74i)16-s + (3.97 + 2.29i)17-s + ⋯
L(s)  = 1  + (−0.471 − 0.126i)2-s + (−0.288 − 0.5i)3-s + (−0.659 − 0.380i)4-s + (1.37 − 1.37i)5-s + (0.0729 + 0.272i)6-s + (−0.739 + 0.198i)7-s + (0.607 + 0.607i)8-s + (−0.166 + 0.288i)9-s + (−0.824 + 0.475i)10-s + (0.00739 − 0.0276i)11-s + 0.439i·12-s + (0.926 + 0.375i)13-s + 0.373·14-s + (−1.08 − 0.291i)15-s + (0.171 + 0.296i)16-s + (0.234 + 0.135i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.112 + 0.993i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.112 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $0.112 + 0.993i$
Analytic conductor: \(1.06267\)
Root analytic conductor: \(1.03086\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :1),\ 0.112 + 0.993i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.585808 - 0.523268i\)
\(L(\frac12)\) \(\approx\) \(0.585808 - 0.523268i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 + 1.5i)T \)
13 \( 1 + (-12.0 - 4.88i)T \)
good2 \( 1 + (0.942 + 0.252i)T + (3.46 + 2i)T^{2} \)
5 \( 1 + (-6.89 + 6.89i)T - 25iT^{2} \)
7 \( 1 + (5.17 - 1.38i)T + (42.4 - 24.5i)T^{2} \)
11 \( 1 + (-0.0813 + 0.303i)T + (-104. - 60.5i)T^{2} \)
17 \( 1 + (-3.97 - 2.29i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-1.10 - 4.13i)T + (-312. + 180.5i)T^{2} \)
23 \( 1 + (-23.2 + 13.4i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-7.34 - 12.7i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (33.3 - 33.3i)T - 961iT^{2} \)
37 \( 1 + (6.45 - 24.0i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (33.5 + 8.97i)T + (1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (-32.3 - 18.6i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-15.4 - 15.4i)T + 2.20e3iT^{2} \)
53 \( 1 - 10.8T + 2.80e3T^{2} \)
59 \( 1 + (36.6 - 9.81i)T + (3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (-14.9 + 25.8i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (86.6 + 23.2i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + (10.0 + 37.6i)T + (-4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (-74.0 - 74.0i)T + 5.32e3iT^{2} \)
79 \( 1 + 61.9T + 6.24e3T^{2} \)
83 \( 1 + (-63.4 + 63.4i)T - 6.88e3iT^{2} \)
89 \( 1 + (16.7 - 62.6i)T + (-6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (6.66 + 24.8i)T + (-8.14e3 + 4.70e3i)T^{2} \)
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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

−16.26964715600636755414239215550, −14.14734929491491428862244244864, −13.26184152344103007052081007840, −12.52938729688123657652005499115, −10.53288163314803282315274555605, −9.311883111984017137154787254602, −8.619715604801391969109465026785, −6.20525735998895922774384600646, −5.05960356385943429873720428715, −1.34152507426206606900294124968, 3.38398961740180950799276163085, 5.76881413116080336673056391164, 7.12253901047791981646273965662, 9.181075797776636355817900252467, 10.00357118230094106801199659535, 10.96006778732334409557846450331, 13.09510052503775354492198520101, 13.82773428213477066777248789488, 15.15837875894840838909662875013, 16.58260298316355002468573034551

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