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Properties

Label 2-390-65.33-c1-0-5
Degree $2$
Conductor $390$
Sign $0.404 - 0.914i$
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 + (0.965 − 0.258i)3-s + (−0.499 − 0.866i)4-s + (2.04 + 0.895i)5-s + (−0.258 + 0.965i)6-s + (−1.39 + 0.804i)7-s + 0.999·8-s + (0.866 − 0.499i)9-s + (−1.80 + 1.32i)10-s + (0.837 + 3.12i)11-s + (−0.707 − 0.707i)12-s + (−0.932 + 3.48i)13-s − 1.60i·14-s + (2.21 + 0.334i)15-s + (−0.5 + 0.866i)16-s + (1.84 − 6.89i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.557 − 0.149i)3-s + (−0.249 − 0.433i)4-s + (0.916 + 0.400i)5-s + (−0.105 + 0.394i)6-s + (−0.526 + 0.304i)7-s + 0.353·8-s + (0.288 − 0.166i)9-s + (−0.569 + 0.419i)10-s + (0.252 + 0.942i)11-s + (−0.204 − 0.204i)12-s + (−0.258 + 0.965i)13-s − 0.430i·14-s + (0.570 + 0.0864i)15-s + (−0.125 + 0.216i)16-s + (0.448 − 1.67i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.404 - 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.404 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.404 - 0.914i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.404 - 0.914i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25336 + 0.816270i\)
\(L(\frac12)\) \(\approx\) \(1.25336 + 0.816270i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.965 + 0.258i)T \)
5 \( 1 + (-2.04 - 0.895i)T \)
13 \( 1 + (0.932 - 3.48i)T \)
good7 \( 1 + (1.39 - 0.804i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.837 - 3.12i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.84 + 6.89i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-5.04 - 1.35i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.676 - 2.52i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (4.32 + 2.49i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.38 - 3.38i)T + 31iT^{2} \)
37 \( 1 + (8.67 + 5.01i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.37 + 0.904i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-4.42 - 1.18i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + 0.302iT - 47T^{2} \)
53 \( 1 + (6.71 + 6.71i)T + 53iT^{2} \)
59 \( 1 + (-1.67 + 6.26i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (0.840 + 1.45i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.61 - 9.72i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.465 + 1.73i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + 9.67T + 73T^{2} \)
79 \( 1 + 11.7iT - 79T^{2} \)
83 \( 1 + 13.9iT - 83T^{2} \)
89 \( 1 + (-13.4 + 3.61i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (3.45 + 5.98i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.49434358292489559448737200224, −10.03671050537741172698990082320, −9.507194367285608336863936067864, −9.043956357943693697856039110973, −7.40587351188210005656375749371, −7.06097532833754637434403015956, −5.89635329594638034800014180464, −4.82820851570437298986267433785, −3.13419304039299154465461803420, −1.80711407513613145383222895040, 1.22517581637851314722057054025, 2.81031149208282597686772175763, 3.75385448028534102785744594116, 5.27847159171510900694629849820, 6.31491995905029061982906923017, 7.74277220277345993017527092795, 8.609022679016256268712747199392, 9.397438792484037987106032346986, 10.22119027735504414074870760476, 10.78238362866384140392856717585

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