Properties

Label 2-390-65.18-c1-0-4
Degree $2$
Conductor $390$
Sign $0.660 - 0.750i$
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  + i·2-s + (−0.707 − 0.707i)3-s − 4-s + (−2.20 − 0.383i)5-s + (0.707 − 0.707i)6-s + 1.52·7-s i·8-s + 1.00i·9-s + (0.383 − 2.20i)10-s + (2.70 + 2.70i)11-s + (0.707 + 0.707i)12-s + (3.17 − 1.71i)13-s + 1.52i·14-s + (1.28 + 1.82i)15-s + 16-s + (4.03 + 4.03i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.408 − 0.408i)3-s − 0.5·4-s + (−0.985 − 0.171i)5-s + (0.288 − 0.288i)6-s + 0.577·7-s − 0.353i·8-s + 0.333i·9-s + (0.121 − 0.696i)10-s + (0.814 + 0.814i)11-s + (0.204 + 0.204i)12-s + (0.879 − 0.475i)13-s + 0.408i·14-s + (0.332 + 0.472i)15-s + 0.250·16-s + (0.978 + 0.978i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.660 - 0.750i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.660 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.660 - 0.750i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.660 - 0.750i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.991332 + 0.448011i\)
\(L(\frac12)\) \(\approx\) \(0.991332 + 0.448011i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (2.20 + 0.383i)T \)
13 \( 1 + (-3.17 + 1.71i)T \)
good7 \( 1 - 1.52T + 7T^{2} \)
11 \( 1 + (-2.70 - 2.70i)T + 11iT^{2} \)
17 \( 1 + (-4.03 - 4.03i)T + 17iT^{2} \)
19 \( 1 + (-1.06 - 1.06i)T + 19iT^{2} \)
23 \( 1 + (0.485 - 0.485i)T - 23iT^{2} \)
29 \( 1 + 2.28iT - 29T^{2} \)
31 \( 1 + (-1 + i)T - 31iT^{2} \)
37 \( 1 - 7.96T + 37T^{2} \)
41 \( 1 + (8.22 - 8.22i)T - 41iT^{2} \)
43 \( 1 + (-5.76 + 5.76i)T - 43iT^{2} \)
47 \( 1 - 2.92T + 47T^{2} \)
53 \( 1 + (3.69 + 3.69i)T + 53iT^{2} \)
59 \( 1 + (4.50 - 4.50i)T - 59iT^{2} \)
61 \( 1 + 2.43T + 61T^{2} \)
67 \( 1 + 3.79iT - 67T^{2} \)
71 \( 1 + (-10.7 + 10.7i)T - 71iT^{2} \)
73 \( 1 - 10.3iT - 73T^{2} \)
79 \( 1 - 1.53iT - 79T^{2} \)
83 \( 1 - 7.72T + 83T^{2} \)
89 \( 1 + (7.21 - 7.21i)T - 89iT^{2} \)
97 \( 1 + 6.36iT - 97T^{2} \)
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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

−11.59597263948071217485749160635, −10.63824927749842883652795224776, −9.469139079958050127905349327664, −8.167957189204590722567790134966, −7.88092089084429009142298227097, −6.74746040110915829303932908005, −5.79413844886344163489203155324, −4.62749688937237017805777513928, −3.65711472819625609184836429060, −1.26756815002001218473645312769, 1.01536066822958718458185143223, 3.16913460468694797374956219007, 4.04119929337578284114204951416, 5.06116165573846804904157193345, 6.33315598209328840734421877180, 7.60890866389377079699975483252, 8.610579146157916953762469722702, 9.402009584626788616484537685851, 10.60565865898598804345940355475, 11.40284400326711143150466513400

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