Properties

Label 2-693-11.4-c1-0-11
Degree $2$
Conductor $693$
Sign $0.132 + 0.991i$
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  + (−1.18 − 0.862i)2-s + (0.0467 + 0.143i)4-s + (0.377 − 0.274i)5-s + (−0.309 − 0.951i)7-s + (−0.837 + 2.57i)8-s − 0.684·10-s + (2.22 + 2.45i)11-s + (1.28 + 0.930i)13-s + (−0.453 + 1.39i)14-s + (3.46 − 2.51i)16-s + (4.22 − 3.07i)17-s + (1.30 − 4.01i)19-s + (0.0571 + 0.0415i)20-s + (−0.527 − 4.83i)22-s + 1.80·23-s + ⋯
L(s)  = 1  + (−0.839 − 0.609i)2-s + (0.0233 + 0.0719i)4-s + (0.168 − 0.122i)5-s + (−0.116 − 0.359i)7-s + (−0.296 + 0.911i)8-s − 0.216·10-s + (0.672 + 0.740i)11-s + (0.355 + 0.257i)13-s + (−0.121 + 0.372i)14-s + (0.865 − 0.628i)16-s + (1.02 − 0.745i)17-s + (0.299 − 0.921i)19-s + (0.0127 + 0.00928i)20-s + (−0.112 − 1.03i)22-s + 0.376·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.132 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.132 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.132 + 0.991i$
Analytic conductor: \(5.53363\)
Root analytic conductor: \(2.35236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (631, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :1/2),\ 0.132 + 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.736017 - 0.644459i\)
\(L(\frac12)\) \(\approx\) \(0.736017 - 0.644459i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (-2.22 - 2.45i)T \)
good2 \( 1 + (1.18 + 0.862i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (-0.377 + 0.274i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (-1.28 - 0.930i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-4.22 + 3.07i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.30 + 4.01i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 1.80T + 23T^{2} \)
29 \( 1 + (0.840 + 2.58i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.04 + 0.760i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.600 + 1.84i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.321 + 0.990i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 8.70T + 43T^{2} \)
47 \( 1 + (-1.97 + 6.08i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (10.6 + 7.76i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.65 - 8.18i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-12.3 + 8.95i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 4.67T + 67T^{2} \)
71 \( 1 + (-7.88 + 5.72i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (4.11 + 12.6i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (2.89 + 2.10i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (13.9 - 10.1i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 8.91T + 89T^{2} \)
97 \( 1 + (-2.18 - 1.58i)T + (29.9 + 92.2i)T^{2} \)
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

−10.06829934348492204541818897191, −9.463182995222198131179083953706, −8.953364833486003513589352117200, −7.75336459984004637281039391969, −6.95883812620189689243171546022, −5.72022199917285940627166039159, −4.77673150023256758720644091567, −3.43196522274180202200582549876, −2.06976273724158154702959789911, −0.864739164790193686678107809974, 1.18382856837166344769334047592, 3.11323989260059587960993426695, 4.01375504898033011078168305076, 5.73758177833828652885032305096, 6.23021858893333176599322032287, 7.33687728023047836314307826530, 8.189779011371665425912247061133, 8.762643325612754441189173954799, 9.662776273160589423672069451113, 10.34923791638664740862171333626

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