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Properties

Label 2-693-7.4-c1-0-6
Degree $2$
Conductor $693$
Sign $-0.815 + 0.578i$
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  + (1.29 + 2.24i)2-s + (−2.36 + 4.09i)4-s + (1.09 + 1.89i)5-s + (−2.61 + 0.413i)7-s − 7.08·8-s + (−2.84 + 4.92i)10-s + (0.5 − 0.866i)11-s − 2.95·13-s + (−4.31 − 5.33i)14-s + (−4.46 − 7.72i)16-s + (1.29 − 2.24i)17-s + (1.17 + 2.04i)19-s − 10.3·20-s + 2.59·22-s + (4.46 + 7.72i)23-s + ⋯
L(s)  = 1  + (0.917 + 1.58i)2-s + (−1.18 + 2.04i)4-s + (0.490 + 0.848i)5-s + (−0.987 + 0.156i)7-s − 2.50·8-s + (−0.899 + 1.55i)10-s + (0.150 − 0.261i)11-s − 0.818·13-s + (−1.15 − 1.42i)14-s + (−1.11 − 1.93i)16-s + (0.314 − 0.544i)17-s + (0.270 + 0.468i)19-s − 2.31·20-s + 0.553·22-s + (0.930 + 1.61i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.815 + 0.578i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.815 + 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $-0.815 + 0.578i$
Analytic conductor: \(5.53363\)
Root analytic conductor: \(2.35236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (298, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :1/2),\ -0.815 + 0.578i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.569755 - 1.78779i\)
\(L(\frac12)\) \(\approx\) \(0.569755 - 1.78779i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.61 - 0.413i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-1.29 - 2.24i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.09 - 1.89i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + 2.95T + 13T^{2} \)
17 \( 1 + (-1.29 + 2.24i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.17 - 2.04i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.46 - 7.72i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.48T + 29T^{2} \)
31 \( 1 + (-0.865 + 1.49i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.19 - 3.79i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 9.68T + 41T^{2} \)
43 \( 1 + 10.3T + 43T^{2} \)
47 \( 1 + (-5.55 - 9.62i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.58 - 7.94i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.46 + 7.72i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.73 + 11.6i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.62 + 2.81i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.994T + 71T^{2} \)
73 \( 1 + (0.147 - 0.255i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.53 - 13.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.26T + 83T^{2} \)
89 \( 1 + (1.30 + 2.25i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 7.51T + 97T^{2} \)
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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

−11.06383085916456126064505790786, −9.686596257936554796699996686546, −9.299552374184382703917771939853, −7.86999593116003103591957993155, −7.25023189956459204974608353833, −6.45824366490221777224281791996, −5.81872735610016835797369544159, −4.94706172975112791377669560040, −3.59707213595806845626712475021, −2.85251497530481924436697255324, 0.73452710140119737190501348920, 2.13917625700991413601044576242, 3.14790597486312893291470946841, 4.25848363323680063175820504572, 5.05897285897794101405583371513, 5.91344288797634995186427277306, 7.08569710381581654794607710039, 8.808225942907691856652947432690, 9.407794452668053427784568781218, 10.13264124127561685375380822005

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