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Properties

Label 2-390-13.9-c1-0-2
Degree $2$
Conductor $390$
Sign $0.859 - 0.511i$
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 + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + 5-s + (0.499 + 0.866i)6-s + (1.5 + 2.59i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)10-s + (−0.5 + 0.866i)11-s − 0.999·12-s + (2.5 − 2.59i)13-s − 3·14-s + (0.5 − 0.866i)15-s + (−0.5 + 0.866i)16-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + 0.447·5-s + (0.204 + 0.353i)6-s + (0.566 + 0.981i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.158 + 0.273i)10-s + (−0.150 + 0.261i)11-s − 0.288·12-s + (0.693 − 0.720i)13-s − 0.801·14-s + (0.129 − 0.223i)15-s + (−0.125 + 0.216i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.859 - 0.511i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.859 - 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36722 + 0.375758i\)
\(L(\frac12)\) \(\approx\) \(1.36722 + 0.375758i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 - T \)
13 \( 1 + (-2.5 + 2.59i)T \)
good7 \( 1 + (-1.5 - 2.59i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 10T + 31T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3 - 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1 - 1.73i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 9T + 47T^{2} \)
53 \( 1 + 13T + 53T^{2} \)
59 \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6 + 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1 - 1.73i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 16T + 73T^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (0.5 - 0.866i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6 + 10.3i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.40655522443016623374508753724, −10.27043887772521701990008525581, −9.401552158342165510545022240980, −8.322000406059020873032228095327, −7.993971716072043059200473055327, −6.58710689688921022555872509310, −5.83154900473575940787590018557, −4.83044610181464682938803592506, −2.94655593616989399592448213886, −1.50504020283557303662799104360, 1.33862244464039955174558479217, 2.94043778047944064439256348064, 4.13057929401455010479128368689, 5.07785596282092560504841060970, 6.62330776080705882361830139222, 7.73415438426780140686718026043, 8.687890017258664319130095368439, 9.515933477543360842369224289040, 10.34764191178301719309077505078, 11.12404713016780228056349221395

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