↑

Properties

Label 2-390-65.64-c1-0-0
Degree $2$
Conductor $390$
Sign $-0.974 - 0.224i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + i·3-s + 4-s + (−0.311 + 2.21i)5-s − i·6-s − 1.52·7-s − 8-s − 9-s + (0.311 − 2.21i)10-s + 2.42i·11-s + i·12-s + (−3.59 − 0.311i)13-s + 1.52·14-s + (−2.21 − 0.311i)15-s + 16-s + 0.903i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577i·3-s + 0.5·4-s + (−0.139 + 0.990i)5-s − 0.408i·6-s − 0.576·7-s − 0.353·8-s − 0.333·9-s + (0.0983 − 0.700i)10-s + 0.732i·11-s + 0.288i·12-s + (−0.996 − 0.0862i)13-s + 0.407·14-s + (−0.571 − 0.0803i)15-s + 0.250·16-s + 0.219i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 - 0.224i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.974 - 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.974 - 0.224i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ -0.974 - 0.224i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0552345 + 0.486767i\)
\(L(\frac12)\) \(\approx\) \(0.0552345 + 0.486767i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - iT \)
5 \( 1 + (0.311 - 2.21i)T \)
13 \( 1 + (3.59 + 0.311i)T \)
good7 \( 1 + 1.52T + 7T^{2} \)
11 \( 1 - 2.42iT - 11T^{2} \)
17 \( 1 - 0.903iT - 17T^{2} \)
19 \( 1 + 4.90iT - 19T^{2} \)
23 \( 1 - 2.90iT - 23T^{2} \)
29 \( 1 + 5.95T + 29T^{2} \)
31 \( 1 - 0.755iT - 31T^{2} \)
37 \( 1 + 0.428T + 37T^{2} \)
41 \( 1 - 7.80iT - 41T^{2} \)
43 \( 1 - 7.18iT - 43T^{2} \)
47 \( 1 + 0.949T + 47T^{2} \)
53 \( 1 - 2.56iT - 53T^{2} \)
59 \( 1 + 2.13iT - 59T^{2} \)
61 \( 1 + 5.05T + 61T^{2} \)
67 \( 1 - 5.18T + 67T^{2} \)
71 \( 1 - 5.37iT - 71T^{2} \)
73 \( 1 - 8.76T + 73T^{2} \)
79 \( 1 - 15.0T + 79T^{2} \)
83 \( 1 - 13.2T + 83T^{2} \)
89 \( 1 - 8.62iT - 89T^{2} \)
97 \( 1 - 11.1T + 97T^{2} \)
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

−11.38761437571835627613438112075, −10.73899779575062291013597202537, −9.654322111597419641636065151057, −9.502219370039987321446551987767, −7.955016662550648835332752390336, −7.14197615977322238462671665506, −6.29525143779207504216013199398, −4.89091823680772441049149018717, −3.45406581769046821272458414643, −2.38392006250036269772555952355, 0.38129261353468290071053407799, 2.02713770982913507663827332085, 3.61374111508433110182600707394, 5.23035589700015478233555688959, 6.21241406943998267554991789679, 7.35346773370490943746002747785, 8.147896596058212607576067919413, 9.020486508408856503705253508802, 9.771652127636244532923900665696, 10.84278737086615805304156125311

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