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Properties

Label 2-693-11.9-c1-0-29
Degree $2$
Conductor $693$
Sign $-0.162 - 0.986i$
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  + (0.804 − 2.47i)2-s + (−3.86 − 2.80i)4-s + (−1.33 − 4.10i)5-s + (−0.809 − 0.587i)7-s + (−5.85 + 4.25i)8-s − 11.2·10-s + (3.29 − 0.388i)11-s + (−0.761 + 2.34i)13-s + (−2.10 + 1.53i)14-s + (2.87 + 8.83i)16-s + (−0.366 − 1.12i)17-s + (2.77 − 2.01i)19-s + (−6.37 + 19.6i)20-s + (1.68 − 8.46i)22-s + 8.32·23-s + ⋯
L(s)  = 1  + (0.568 − 1.75i)2-s + (−1.93 − 1.40i)4-s + (−0.596 − 1.83i)5-s + (−0.305 − 0.222i)7-s + (−2.07 + 1.50i)8-s − 3.55·10-s + (0.993 − 0.117i)11-s + (−0.211 + 0.650i)13-s + (−0.563 + 0.409i)14-s + (0.717 + 2.20i)16-s + (−0.0889 − 0.273i)17-s + (0.636 − 0.462i)19-s + (−1.42 + 4.39i)20-s + (0.359 − 1.80i)22-s + 1.73·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.162 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.162 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $-0.162 - 0.986i$
Analytic conductor: \(5.53363\)
Root analytic conductor: \(2.35236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :1/2),\ -0.162 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.873656 + 1.02924i\)
\(L(\frac12)\) \(\approx\) \(0.873656 + 1.02924i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.29 + 0.388i)T \)
good2 \( 1 + (-0.804 + 2.47i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (1.33 + 4.10i)T + (-4.04 + 2.93i)T^{2} \)
13 \( 1 + (0.761 - 2.34i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.366 + 1.12i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-2.77 + 2.01i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 8.32T + 23T^{2} \)
29 \( 1 + (1.17 + 0.854i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.923 + 2.84i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (5.13 + 3.72i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-4.98 + 3.62i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 3.15T + 43T^{2} \)
47 \( 1 + (7.06 - 5.13i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.06 - 6.35i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (9.50 + 6.90i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-2.40 - 7.38i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 3.11T + 67T^{2} \)
71 \( 1 + (-1.69 - 5.21i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (7.00 + 5.09i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (2.73 - 8.40i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (2.98 + 9.19i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 0.985T + 89T^{2} \)
97 \( 1 + (-2.15 + 6.63i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.745181522955589427964689248206, −9.168724821070040709919387068457, −8.773457121171843463979976883599, −7.29987438146650366982118759693, −5.67793175943810216564874235045, −4.70044612207417172962363787052, −4.24028352247832707949334450789, −3.19822732763910043878984144390, −1.57982284652559106778255384251, −0.64532102266314843632841305352, 3.10195693787056656562348630601, 3.68185878916930515008780015951, 4.95744247776345750322343863098, 6.10131169711836180808212835981, 6.71657181524940521295231280102, 7.28171552186044189657257663848, 8.044166516751228666767784172127, 9.078901144215426381563792438726, 10.08880763243982295392872010459, 11.12647041234019325791274065845

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