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Properties

Label 2-693-11.5-c1-0-23
Degree $2$
Conductor $693$
Sign $0.943 + 0.331i$
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  + (0.405 + 1.24i)2-s + (0.223 − 0.162i)4-s + (1.06 − 3.29i)5-s + (−0.809 + 0.587i)7-s + (2.41 + 1.75i)8-s + 4.54·10-s + (0.955 − 3.17i)11-s + (−1.71 − 5.26i)13-s + (−1.06 − 0.771i)14-s + (−1.04 + 3.20i)16-s + (−1.53 + 4.72i)17-s + (−6.65 − 4.83i)19-s + (−0.295 − 0.909i)20-s + (4.35 − 0.0954i)22-s + 7.56·23-s + ⋯
L(s)  = 1  + (0.286 + 0.882i)2-s + (0.111 − 0.0812i)4-s + (0.478 − 1.47i)5-s + (−0.305 + 0.222i)7-s + (0.854 + 0.621i)8-s + 1.43·10-s + (0.288 − 0.957i)11-s + (−0.474 − 1.46i)13-s + (−0.283 − 0.206i)14-s + (−0.260 + 0.801i)16-s + (−0.372 + 1.14i)17-s + (−1.52 − 1.10i)19-s + (−0.0660 − 0.203i)20-s + (0.928 − 0.0203i)22-s + 1.57·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.331i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 + 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.943 + 0.331i$
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: \((2,\ 693,\ (\ :1/2),\ 0.943 + 0.331i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.99431 - 0.340272i\)
\(L(\frac12)\) \(\approx\) \(1.99431 - 0.340272i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.955 + 3.17i)T \)
good2 \( 1 + (-0.405 - 1.24i)T + (-1.61 + 1.17i)T^{2} \)
5 \( 1 + (-1.06 + 3.29i)T + (-4.04 - 2.93i)T^{2} \)
13 \( 1 + (1.71 + 5.26i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.53 - 4.72i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (6.65 + 4.83i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 7.56T + 23T^{2} \)
29 \( 1 + (1.18 - 0.864i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.745 - 2.29i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-2.29 + 1.66i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-5.60 - 4.07i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 5.62T + 43T^{2} \)
47 \( 1 + (0.0337 + 0.0245i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-1.38 - 4.25i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (2.42 - 1.76i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (3.60 - 11.0i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 9.95T + 67T^{2} \)
71 \( 1 + (-2.16 + 6.65i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (5.97 - 4.33i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-5.02 - 15.4i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.98 + 9.17i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 7.94T + 89T^{2} \)
97 \( 1 + (0.726 + 2.23i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.55339450117587823614126789462, −9.186646912259705734264979620150, −8.645997080600198705355382733312, −7.87840240228806627115030594626, −6.64160954315244981747081624525, −5.85836821111086018676934984475, −5.22420081107821720483489985108, −4.32709304790698571874430254880, −2.59328070565025667220027256545, −1.01025231825328478071755168291, 1.98090844141100382439817257006, 2.59492976631168346417931285972, 3.78420455697972677580393192220, 4.64874046754470394931975916940, 6.41574643321248613605415936364, 6.88257663933801469714340647562, 7.54267212888782540729263516899, 9.274827813012939443471297030187, 9.832464657906098279005306026571, 10.72581478704018444661727927014

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