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Properties

Label 2-693-11.3-c1-0-5
Degree $2$
Conductor $693$
Sign $-0.993 - 0.114i$
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  + (−2.14 + 1.55i)2-s + (1.55 − 4.78i)4-s + (1.38 + 1.00i)5-s + (−0.309 + 0.951i)7-s + (2.48 + 7.64i)8-s − 4.54·10-s + (−2.49 − 2.18i)11-s + (−1.22 + 0.888i)13-s + (−0.819 − 2.52i)14-s + (−9.11 − 6.62i)16-s + (−1.93 − 1.40i)17-s + (1.95 + 6.02i)19-s + (6.97 − 5.06i)20-s + (8.75 + 0.795i)22-s + 5.05·23-s + ⋯
L(s)  = 1  + (−1.51 + 1.10i)2-s + (0.777 − 2.39i)4-s + (0.619 + 0.450i)5-s + (−0.116 + 0.359i)7-s + (0.878 + 2.70i)8-s − 1.43·10-s + (−0.752 − 0.658i)11-s + (−0.339 + 0.246i)13-s + (−0.219 − 0.674i)14-s + (−2.27 − 1.65i)16-s + (−0.469 − 0.341i)17-s + (0.448 + 1.38i)19-s + (1.55 − 1.13i)20-s + (1.86 + 0.169i)22-s + 1.05·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.114i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 - 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $-0.993 - 0.114i$
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: \((2,\ 693,\ (\ :1/2),\ -0.993 - 0.114i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0277048 + 0.484239i\)
\(L(\frac12)\) \(\approx\) \(0.0277048 + 0.484239i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.49 + 2.18i)T \)
good2 \( 1 + (2.14 - 1.55i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (-1.38 - 1.00i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (1.22 - 0.888i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.93 + 1.40i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.95 - 6.02i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 5.05T + 23T^{2} \)
29 \( 1 + (2.57 - 7.93i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (4.88 - 3.54i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.30 - 7.10i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.55 - 4.78i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 2.02T + 43T^{2} \)
47 \( 1 + (-1.70 - 5.25i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-5.04 + 3.66i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.45 - 10.6i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (11.1 + 8.11i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 13.6T + 67T^{2} \)
71 \( 1 + (11.5 + 8.42i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.441 - 1.36i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (5.43 - 3.94i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-4.74 - 3.45i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 16.4T + 89T^{2} \)
97 \( 1 + (9.74 - 7.08i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.56957304148033285749847602883, −9.790096196786967060379435814232, −9.082543208629966329201654054223, −8.331085875664552634015803808735, −7.43749536499936668432399915304, −6.67160343577078547522785343707, −5.82538681281014331393847203639, −5.12586466950483169183236472353, −2.90171120147881524531530674755, −1.52630953184281189018113277747, 0.41648676554171505217232410654, 1.90740630070246445279439066853, 2.77195769962929199992623669793, 4.16170152229583985852166384179, 5.47740694391306011887442459674, 7.12346215186847059051398192834, 7.54576265105133909252225010186, 8.728722856495720963202892042694, 9.349943149260382251145019524893, 9.917892359042027241805552815047

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