Properties

Label 2-690-15.2-c1-0-25
Degree $2$
Conductor $690$
Sign $0.865 + 0.501i$
Analytic cond. $5.50967$
Root an. cond. $2.34727$
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  + (−0.707 + 0.707i)2-s + (1.70 − 0.292i)3-s − 1.00i·4-s + (−2.12 + 0.707i)5-s + (−0.999 + 1.41i)6-s + (−1 − i)7-s + (0.707 + 0.707i)8-s + (2.82 − i)9-s + (0.999 − 2i)10-s − 1.41i·11-s + (−0.292 − 1.70i)12-s + (2 − 2i)13-s + 1.41·14-s + (−3.41 + 1.82i)15-s − 1.00·16-s + (1.41 − 1.41i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.985 − 0.169i)3-s − 0.500i·4-s + (−0.948 + 0.316i)5-s + (−0.408 + 0.577i)6-s + (−0.377 − 0.377i)7-s + (0.250 + 0.250i)8-s + (0.942 − 0.333i)9-s + (0.316 − 0.632i)10-s − 0.426i·11-s + (−0.0845 − 0.492i)12-s + (0.554 − 0.554i)13-s + 0.377·14-s + (−0.881 + 0.472i)15-s − 0.250·16-s + (0.342 − 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 + 0.501i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.865 + 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.865 + 0.501i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ 0.865 + 0.501i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27481 - 0.342920i\)
\(L(\frac12)\) \(\approx\) \(1.27481 - 0.342920i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (-1.70 + 0.292i)T \)
5 \( 1 + (2.12 - 0.707i)T \)
23 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 + (1 + i)T + 7iT^{2} \)
11 \( 1 + 1.41iT - 11T^{2} \)
13 \( 1 + (-2 + 2i)T - 13iT^{2} \)
17 \( 1 + (-1.41 + 1.41i)T - 17iT^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
29 \( 1 - 4.24T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + (-6 - 6i)T + 37iT^{2} \)
41 \( 1 + 5.65iT - 41T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + (2.82 - 2.82i)T - 47iT^{2} \)
53 \( 1 + (8.48 + 8.48i)T + 53iT^{2} \)
59 \( 1 - 1.41T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + (2 + 2i)T + 67iT^{2} \)
71 \( 1 - 5.65iT - 71T^{2} \)
73 \( 1 + (1 - i)T - 73iT^{2} \)
79 \( 1 + 2iT - 79T^{2} \)
83 \( 1 + (-8.48 - 8.48i)T + 83iT^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (-9 - 9i)T + 97iT^{2} \)
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

−10.23323184539171176030455292781, −9.379918970053716101588081163262, −8.393344307537646212548362917667, −7.977131460593409579295938594143, −7.03890864584422895233984119923, −6.38228307976293236413924556091, −4.80370220038786041116755867346, −3.65630461909943278392885957277, −2.77704326513832913268829726790, −0.817567009507930712467715354000, 1.48773605658242442433617504051, 2.87982686534289537442444609591, 3.81268517560746395204398509980, 4.59845458988797496856352537905, 6.27989449214908656173478994448, 7.50685730904725941946415915664, 8.075782157929036270227928590208, 8.869432139296148415266535458222, 9.544485603357143191324189534082, 10.38334829633922446549281305387

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