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Properties

Label 2-390-65.4-c1-0-0
Degree $2$
Conductor $390$
Sign $-0.999 - 0.0431i$
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 + (2.20 − 0.341i)10-s + (−1.14 − 0.658i)11-s + 0.999i·12-s + (−2.41 + 2.67i)13-s − 1.52·14-s + (0.341 + 2.20i)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 + (0.698 − 0.107i)10-s + (−0.343 − 0.198i)11-s + 0.288i·12-s + (−0.669 + 0.743i)13-s − 0.408·14-s + (0.0881 + 0.570i)15-s + (−0.125 + 0.216i)16-s + (−0.329 + 0.190i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0431i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0431i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.999 - 0.0431i$
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: \((2,\ 390,\ (\ :1/2),\ -0.999 - 0.0431i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00260065 + 0.120464i\)
\(L(\frac12)\) \(\approx\) \(0.00260065 + 0.120464i\)
\(L(\frac{3}{2})\) 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

−11.90271413797703094440347556638, −10.86613220417268427493445178046, −9.852070351727496076910610868890, −8.668606292457108568831127969564, −8.192364334362207975955397008510, −7.11493810122828457372209774057, −6.08951766644698392897000698343, −5.07956772086142877876269106962, −4.17498601084046590795204903081, −1.90799401330257764817075776668, 0.090665345169537599148485362180, 2.36830862471692709465186895584, 3.74696942820947032660676397892, 4.65734861543605155150689208007, 6.08033855743982567999917182076, 7.38642100872948476753915520279, 7.906768255879227016561258004904, 9.281731652915662482056542428090, 10.35038391903440742036432114115, 10.69828379023047902005591499702

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