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Properties

Label 2-390-195.149-c1-0-8
Degree $2$
Conductor $390$
Sign $0.994 - 0.108i$
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  + (−0.258 − 0.965i)2-s + (−1.38 + 1.03i)3-s + (−0.866 + 0.499i)4-s + (2.16 − 0.543i)5-s + (1.36 + 1.07i)6-s + (−0.476 − 0.127i)7-s + (0.707 + 0.707i)8-s + (0.849 − 2.87i)9-s + (−1.08 − 1.95i)10-s + (1.42 + 5.33i)11-s + (0.683 − 1.59i)12-s + (−3.60 + 0.165i)13-s + 0.492i·14-s + (−2.44 + 3.00i)15-s + (0.500 − 0.866i)16-s + (2.57 − 1.48i)17-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (−0.801 + 0.598i)3-s + (−0.433 + 0.249i)4-s + (0.970 − 0.243i)5-s + (0.555 + 0.437i)6-s + (−0.179 − 0.0482i)7-s + (0.249 + 0.249i)8-s + (0.283 − 0.959i)9-s + (−0.343 − 0.618i)10-s + (0.430 + 1.60i)11-s + (0.197 − 0.459i)12-s + (−0.998 + 0.0458i)13-s + 0.131i·14-s + (−0.631 + 0.775i)15-s + (0.125 − 0.216i)16-s + (0.624 − 0.360i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.108i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.994 - 0.108i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.994 - 0.108i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07355 + 0.0586378i\)
\(L(\frac12)\) \(\approx\) \(1.07355 + 0.0586378i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
3 \( 1 + (1.38 - 1.03i)T \)
5 \( 1 + (-2.16 + 0.543i)T \)
13 \( 1 + (3.60 - 0.165i)T \)
good7 \( 1 + (0.476 + 0.127i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-1.42 - 5.33i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-2.57 + 1.48i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.31 - 1.15i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-5.06 - 2.92i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.83 - 2.79i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.68 + 2.68i)T - 31iT^{2} \)
37 \( 1 + (-1.20 - 4.49i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (0.293 - 0.0785i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-0.151 - 0.262i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.68 - 4.68i)T + 47iT^{2} \)
53 \( 1 + 8.82T + 53T^{2} \)
59 \( 1 + (12.8 + 3.45i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.638 - 1.10i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.45 + 0.927i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-3.67 + 13.7i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-1.84 + 1.84i)T - 73iT^{2} \)
79 \( 1 + 5.77T + 79T^{2} \)
83 \( 1 + (7.09 - 7.09i)T - 83iT^{2} \)
89 \( 1 + (-2.33 - 8.69i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-2.93 + 10.9i)T + (-84.0 - 48.5i)T^{2} \)
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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

−11.30789708510220046662898261679, −10.15649934267767826829786484284, −9.706761248827544096669655748921, −9.252953358750012757137147743929, −7.51143088067954408580536825763, −6.49566706949605590975393797142, −5.14527491913913235908566092934, −4.64643377222754367933832190221, −3.02244535799651872300456227334, −1.39632890532550568300473305364, 1.01054715068533957300453818152, 2.90125116279791083793007706869, 4.92020867131953012805277928919, 5.76904943699753827963611494615, 6.44098306814728405506800848553, 7.31744457415959113406364971994, 8.425308844152446003573334376889, 9.467591182081287418378137432193, 10.37296811588093785796131559199, 11.20982458183572470720944993269

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