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Properties

Label 2-39-39.8-c1-0-1
Degree $2$
Conductor $39$
Sign $0.789 + 0.614i$
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  + (0.707 − 0.707i)2-s + (−1 − 1.41i)3-s + 0.999i·4-s + (−1.41 + 1.41i)5-s + (−1.70 − 0.292i)6-s + (1 − i)7-s + (2.12 + 2.12i)8-s + (−1.00 + 2.82i)9-s + 2.00i·10-s + (−2.82 − 2.82i)11-s + (1.41 − 0.999i)12-s + (−2 − 3i)13-s − 1.41i·14-s + (3.41 + 0.585i)15-s + 1.00·16-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.577 − 0.816i)3-s + 0.499i·4-s + (−0.632 + 0.632i)5-s + (−0.696 − 0.119i)6-s + (0.377 − 0.377i)7-s + (0.750 + 0.750i)8-s + (−0.333 + 0.942i)9-s + 0.632i·10-s + (−0.852 − 0.852i)11-s + (0.408 − 0.288i)12-s + (−0.554 − 0.832i)13-s − 0.377i·14-s + (0.881 + 0.151i)15-s + 0.250·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.789 + 0.614i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.789 + 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $0.789 + 0.614i$
Analytic conductor: \(0.311416\)
Root analytic conductor: \(0.558047\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :1/2),\ 0.789 + 0.614i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.744802 - 0.255660i\)
\(L(\frac12)\) \(\approx\) \(0.744802 - 0.255660i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1 + 1.41i)T \)
13 \( 1 + (2 + 3i)T \)
good2 \( 1 + (-0.707 + 0.707i)T - 2iT^{2} \)
5 \( 1 + (1.41 - 1.41i)T - 5iT^{2} \)
7 \( 1 + (-1 + i)T - 7iT^{2} \)
11 \( 1 + (2.82 + 2.82i)T + 11iT^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + (-1 - i)T + 19iT^{2} \)
23 \( 1 - 8.48T + 23T^{2} \)
29 \( 1 - 2.82iT - 29T^{2} \)
31 \( 1 + (5 + 5i)T + 31iT^{2} \)
37 \( 1 + (-1 + i)T - 37iT^{2} \)
41 \( 1 + (1.41 - 1.41i)T - 41iT^{2} \)
43 \( 1 - 6iT - 43T^{2} \)
47 \( 1 + (2.82 + 2.82i)T + 47iT^{2} \)
53 \( 1 + 5.65iT - 53T^{2} \)
59 \( 1 + (2.82 + 2.82i)T + 59iT^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 + (5 + 5i)T + 67iT^{2} \)
71 \( 1 + (-2.82 + 2.82i)T - 71iT^{2} \)
73 \( 1 + (-1 + i)T - 73iT^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 + (5.65 - 5.65i)T - 83iT^{2} \)
89 \( 1 + (-9.89 - 9.89i)T + 89iT^{2} \)
97 \( 1 + (-7 - 7i)T + 97iT^{2} \)
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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

−16.37559412782342773339728704424, −14.76355965732462353629494019899, −13.41064780569735653019484224142, −12.66745709481539665334697582035, −11.30675612783787748154691379910, −10.85600526130536181306413773038, −8.061664789178817862340467456829, −7.27446410265545340213486625433, −5.19091605701394411623288195776, −3.05896449183021981084934127189, 4.53048237557237169882638974755, 5.27301574152984749826579822970, 7.09147578880902535888251385419, 9.046220776353028334574936459101, 10.33856833734465824807928151464, 11.63744982230134169591864503918, 12.83886897395357891634795526795, 14.52845314024573689676075078205, 15.35277596935437032890438728584, 16.08258772382755486974181945960

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