Properties

Label 2-390-5.4-c1-0-8
Degree $2$
Conductor $390$
Sign $-0.447 + 0.894i$
Analytic cond. $3.11416$
Root an. cond. $1.76469$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s + i·3-s − 4-s + (−2 − i)5-s − 6-s i·8-s − 9-s + (1 − 2i)10-s − 6·11-s i·12-s i·13-s + (1 − 2i)15-s + 16-s i·18-s − 6·19-s + (2 + i)20-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (−0.894 − 0.447i)5-s − 0.408·6-s − 0.353i·8-s − 0.333·9-s + (0.316 − 0.632i)10-s − 1.80·11-s − 0.288i·12-s − 0.277i·13-s + (0.258 − 0.516i)15-s + 0.250·16-s − 0.235i·18-s − 1.37·19-s + (0.447 + 0.223i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
5 \( 1 + (2 + i)T \)
13 \( 1 + iT \)
good7 \( 1 - 7T^{2} \)
11 \( 1 + 6T + 11T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 6T + 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 - 10iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 - 8iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 10T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 + 16T + 79T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.70809765424301575703797741022, −10.27860822179830146277111877190, −8.834150179522979669111286677836, −8.262799860054498916787014023187, −7.47020225606372979519662389201, −6.15728573959655977856396757472, −4.98628006169101036492933345719, −4.36069336101473660250875034031, −2.90566405700736196445289808821, 0, 2.18171372071285739740026906105, 3.28416136422331593769413605176, 4.55528528910644154233824829377, 5.79275595864651550240912883168, 7.16534286090752305271621679780, 7.914469002022591089368447615768, 8.721114826406882882491544725395, 10.09877033016297426394059805825, 10.85240569269252447258421599262

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