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Properties

Label 2-693-11.3-c1-0-9
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

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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: $\chi_{693} (190, \cdot )$
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 \( 1 + (-2.18 + 1.58i)T + (29.9 - 92.2i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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