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Properties

Label 2-693-77.9-c1-0-2
Degree $2$
Conductor $693$
Sign $-0.589 + 0.807i$
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  + (1.82 + 2.02i)2-s + (−0.570 + 5.42i)4-s + (−2.89 − 0.615i)5-s + (−2.43 − 1.03i)7-s + (−7.63 + 5.54i)8-s + (−4.03 − 6.99i)10-s + (0.272 + 3.30i)11-s + (1.13 − 3.50i)13-s + (−2.34 − 6.83i)14-s + (−14.5 − 3.08i)16-s + (−0.929 + 1.03i)17-s + (0.159 + 1.51i)19-s + (4.98 − 15.3i)20-s + (−6.20 + 6.59i)22-s + (−2.58 + 4.48i)23-s + ⋯
L(s)  = 1  + (1.29 + 1.43i)2-s + (−0.285 + 2.71i)4-s + (−1.29 − 0.275i)5-s + (−0.919 − 0.391i)7-s + (−2.69 + 1.96i)8-s + (−1.27 − 2.21i)10-s + (0.0823 + 0.996i)11-s + (0.315 − 0.972i)13-s + (−0.626 − 1.82i)14-s + (−3.63 − 0.771i)16-s + (−0.225 + 0.250i)17-s + (0.0365 + 0.347i)19-s + (1.11 − 3.43i)20-s + (−1.32 + 1.40i)22-s + (−0.539 + 0.934i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.589 + 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.589 + 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $-0.589 + 0.807i$
Analytic conductor: \(5.53363\)
Root analytic conductor: \(2.35236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :1/2),\ -0.589 + 0.807i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.543152 - 1.06830i\)
\(L(\frac12)\) \(\approx\) \(0.543152 - 1.06830i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.43 + 1.03i)T \)
11 \( 1 + (-0.272 - 3.30i)T \)
good2 \( 1 + (-1.82 - 2.02i)T + (-0.209 + 1.98i)T^{2} \)
5 \( 1 + (2.89 + 0.615i)T + (4.56 + 2.03i)T^{2} \)
13 \( 1 + (-1.13 + 3.50i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.929 - 1.03i)T + (-1.77 - 16.9i)T^{2} \)
19 \( 1 + (-0.159 - 1.51i)T + (-18.5 + 3.95i)T^{2} \)
23 \( 1 + (2.58 - 4.48i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.24 - 2.35i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (3.19 - 0.679i)T + (28.3 - 12.6i)T^{2} \)
37 \( 1 + (-4.05 + 1.80i)T + (24.7 - 27.4i)T^{2} \)
41 \( 1 + (6.29 - 4.57i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 5.34T + 43T^{2} \)
47 \( 1 + (-0.799 - 7.60i)T + (-45.9 + 9.77i)T^{2} \)
53 \( 1 + (-2.84 + 0.603i)T + (48.4 - 21.5i)T^{2} \)
59 \( 1 + (0.800 - 7.61i)T + (-57.7 - 12.2i)T^{2} \)
61 \( 1 + (-0.892 - 0.189i)T + (55.7 + 24.8i)T^{2} \)
67 \( 1 + (-5.18 - 8.97i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.30 + 10.1i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.531 - 5.06i)T + (-71.4 - 15.1i)T^{2} \)
79 \( 1 + (-3.79 - 4.21i)T + (-8.25 + 78.5i)T^{2} \)
83 \( 1 + (-0.465 - 1.43i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-5.95 + 10.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.60 + 8.00i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.46639206660248101259923850412, −10.09475883102961392191003419750, −8.843938933326027871196490529293, −7.917242866999757543245889488001, −7.44421096470222072527874798669, −6.61682059091862564324320996959, −5.67676250975783141773095330800, −4.61039913076566315258983823270, −3.86287069909449702136042110081, −3.14813312199130528263605811761, 0.39959241302806289690964217252, 2.32326616793605030121762624507, 3.40716319627919528798096230658, 3.90000975752837234196426224205, 4.94819998968265278980280159406, 6.15973914863990070092227658631, 6.79340555747088643260100975272, 8.457502944178005367192199293265, 9.319989171903303411115198704532, 10.31083293833119119206557288903

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