↑

Properties

Label 2-693-11.5-c1-0-4
Degree $2$
Conductor $693$
Sign $0.879 + 0.475i$
Analytic cond. $5.53363$
Root an. cond. $2.35236$
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  + (−0.643 − 1.98i)2-s + (−1.88 + 1.37i)4-s + (−0.411 + 1.26i)5-s + (−0.809 + 0.587i)7-s + (0.564 + 0.410i)8-s + 2.77·10-s + (1.44 − 2.98i)11-s + (1.77 + 5.46i)13-s + (1.68 + 1.22i)14-s + (−0.994 + 3.05i)16-s + (−1.32 + 4.06i)17-s + (−0.131 − 0.0955i)19-s + (−0.961 − 2.95i)20-s + (−6.84 − 0.937i)22-s − 1.12·23-s + ⋯
L(s)  = 1  + (−0.454 − 1.40i)2-s + (−0.944 + 0.686i)4-s + (−0.184 + 0.566i)5-s + (−0.305 + 0.222i)7-s + (0.199 + 0.145i)8-s + 0.877·10-s + (0.435 − 0.900i)11-s + (0.492 + 1.51i)13-s + (0.450 + 0.327i)14-s + (−0.248 + 0.764i)16-s + (−0.320 + 0.985i)17-s + (−0.0301 − 0.0219i)19-s + (−0.214 − 0.661i)20-s + (−1.45 − 0.199i)22-s − 0.234·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.879 + 0.475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.879 + 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.879 + 0.475i$
Analytic conductor: \(5.53363\)
Root analytic conductor: \(2.35236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :1/2),\ 0.879 + 0.475i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.952027 - 0.241079i\)
\(L(\frac12)\) \(\approx\) \(0.952027 - 0.241079i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 + (-1.44 + 2.98i)T \)
good2 \( 1 + (0.643 + 1.98i)T + (-1.61 + 1.17i)T^{2} \)
5 \( 1 + (0.411 - 1.26i)T + (-4.04 - 2.93i)T^{2} \)
13 \( 1 + (-1.77 - 5.46i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.32 - 4.06i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.131 + 0.0955i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 1.12T + 23T^{2} \)
29 \( 1 + (-7.95 + 5.78i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.81 - 8.64i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-6.81 + 4.95i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-3.64 - 2.64i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 8.02T + 43T^{2} \)
47 \( 1 + (2.26 + 1.64i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-3.24 - 9.97i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (11.0 - 8.00i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-2.49 + 7.67i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 9.10T + 67T^{2} \)
71 \( 1 + (0.314 - 0.969i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-6.18 + 4.49i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (0.00169 + 0.00520i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (0.875 - 2.69i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 10.5T + 89T^{2} \)
97 \( 1 + (-3.75 - 11.5i)T + (-78.4 + 57.0i)T^{2} \)
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

−10.62711467728268107003176564038, −9.655135364394027432111563402010, −8.901440072858140560495461646783, −8.258494262245921131252510979707, −6.65448799753546034074893860695, −6.21306355882746360342068268446, −4.37438178902053507950726367643, −3.52592594770558382550585020978, −2.56361664381097513053658024702, −1.28275508152411588111018322873, 0.69432457237675907844725262448, 2.89212900074251298446322318037, 4.47249499743562161057854265947, 5.25857473329934440657876666482, 6.33170076096454704860290580372, 7.00546932791878187693675870037, 7.995697405846145181287408550094, 8.469509877939830350197977772367, 9.513718323970035402653960202000, 10.09497605894875697441148161579

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