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Properties

Label 2-39-39.8-c1-0-0
Degree $2$
Conductor $39$
Sign $0.315 - 0.948i$
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 + (−0.292 − 1.70i)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 + (0.585 + 3.41i)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 + (−0.119 − 0.696i)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 + (0.151 + 0.881i)15-s + 0.250·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.315 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.315 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $0.315 - 0.948i$
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: \((2,\ 39,\ (\ :1/2),\ 0.315 - 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.468167 + 0.337532i\)
\(L(\frac12)\) \(\approx\) \(0.468167 + 0.337532i\)
\(L(\frac{3}{2})\) 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

−16.78143769831990227594536998304, −15.69877238775338654574605751273, −14.51485899582245118845021311444, −12.78060801443246797948558853327, −11.77109277835793986615228769704, −10.02155482040305598488882210727, −9.211180387486936601493784758346, −7.64074234927874492951923188733, −5.91663455141999524000695565236, −4.20924296412302041216682511933, 2.01645263655581689352851342192, 5.60369323473452934674015467315, 6.70588095140817211226667910466, 8.664988877066061889315090811529, 10.08889150269282229426962884314, 11.27655515850053202120672354428, 12.04534595841437143990557330796, 13.93288389298878883984517220529, 14.46714382280142753892680151742, 16.38478840930367891237437276030

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