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Properties

Label 2-390-65.4-c1-0-9
Degree $2$
Conductor $390$
Sign $0.172 + 0.984i$
Analytic cond. $3.11416$
Root an. cond. $1.76469$
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  + (0.5 − 0.866i)2-s + (0.866 + 0.5i)3-s + (−0.499 − 0.866i)4-s + (1.40 − 1.74i)5-s + (0.866 − 0.499i)6-s + (−0.763 − 1.32i)7-s − 0.999·8-s + (0.499 + 0.866i)9-s + (−0.809 − 2.08i)10-s + (−1.14 − 0.658i)11-s − 0.999i·12-s + (2.41 − 2.67i)13-s − 1.52·14-s + (2.08 − 0.809i)15-s + (−0.5 + 0.866i)16-s + (1.35 − 0.784i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.499 + 0.288i)3-s + (−0.249 − 0.433i)4-s + (0.626 − 0.779i)5-s + (0.353 − 0.204i)6-s + (−0.288 − 0.500i)7-s − 0.353·8-s + (0.166 + 0.288i)9-s + (−0.255 − 0.659i)10-s + (−0.343 − 0.198i)11-s − 0.288i·12-s + (0.669 − 0.743i)13-s − 0.408·14-s + (0.538 − 0.208i)15-s + (−0.125 + 0.216i)16-s + (0.329 − 0.190i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.172 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.172 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.172 + 0.984i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.172 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49815 - 1.25805i\)
\(L(\frac12)\) \(\approx\) \(1.49815 - 1.25805i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 + (-1.40 + 1.74i)T \)
13 \( 1 + (-2.41 + 2.67i)T \)
good7 \( 1 + (0.763 + 1.32i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.14 + 0.658i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.35 + 0.784i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.18 - 2.41i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-7.31 - 4.22i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.21 - 3.83i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.62iT - 31T^{2} \)
37 \( 1 + (-1.40 + 2.42i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.35 - 0.784i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.58 + 2.64i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 4.94T + 47T^{2} \)
53 \( 1 - 13.9iT - 53T^{2} \)
59 \( 1 + (9.07 - 5.23i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.49 - 4.31i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.38 - 2.40i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (12.8 - 7.41i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 5.98T + 73T^{2} \)
79 \( 1 - 4.87T + 79T^{2} \)
83 \( 1 - 6.39T + 83T^{2} \)
89 \( 1 + (-15.9 - 9.22i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.963 + 1.66i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.82538583268635195090391602728, −10.37134505379783508692461156399, −9.300471846460922580118664355397, −8.678206671806998762434670122374, −7.49657962925452430174801601881, −5.99785961380860938923739734368, −5.12684545088078812108894489496, −3.96424101760874028006702336647, −2.87509331469608619608555601740, −1.27770961806971231120329976018, 2.21876209390854507801896441858, 3.29706361628875731134646624939, 4.71392984286043924067825549048, 6.10883713516292726501667384476, 6.61270335631142482531405636562, 7.64456790101158310845119582862, 8.765014516612624855660290433383, 9.430657289630641610751525766526, 10.60300329738062842111294531186, 11.53563759861850945897611672245

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