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Properties

Label 2-693-11.9-c1-0-25
Degree $2$
Conductor $693$
Sign $-0.879 + 0.475i$
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  + (0.643 − 1.98i)2-s + (−1.88 − 1.37i)4-s + (0.411 + 1.26i)5-s + (−0.809 − 0.587i)7-s + (−0.564 + 0.410i)8-s + 2.77·10-s + (−1.44 − 2.98i)11-s + (1.77 − 5.46i)13-s + (−1.68 + 1.22i)14-s + (−0.994 − 3.05i)16-s + (1.32 + 4.06i)17-s + (−0.131 + 0.0955i)19-s + (0.961 − 2.95i)20-s + (−6.84 + 0.937i)22-s + 1.12·23-s + ⋯
L(s)  = 1  + (0.454 − 1.40i)2-s + (−0.944 − 0.686i)4-s + (0.184 + 0.566i)5-s + (−0.305 − 0.222i)7-s + (−0.199 + 0.145i)8-s + 0.877·10-s + (−0.435 − 0.900i)11-s + (0.492 − 1.51i)13-s + (−0.450 + 0.327i)14-s + (−0.248 − 0.764i)16-s + (0.320 + 0.985i)17-s + (−0.0301 + 0.0219i)19-s + (0.214 − 0.661i)20-s + (−1.45 + 0.199i)22-s + 0.234·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.879 + 0.475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.879 + 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $-0.879 + 0.475i$
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: \((2,\ 693,\ (\ :1/2),\ -0.879 + 0.475i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.438534 - 1.73178i\)
\(L(\frac12)\) \(\approx\) \(0.438534 - 1.73178i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.44 + 2.98i)T \)
good2 \( 1 + (-0.643 + 1.98i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (-0.411 - 1.26i)T + (-4.04 + 2.93i)T^{2} \)
13 \( 1 + (-1.77 + 5.46i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.32 - 4.06i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.131 - 0.0955i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 1.12T + 23T^{2} \)
29 \( 1 + (7.95 + 5.78i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.81 + 8.64i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-6.81 - 4.95i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (3.64 - 2.64i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 8.02T + 43T^{2} \)
47 \( 1 + (-2.26 + 1.64i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (3.24 - 9.97i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-11.0 - 8.00i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-2.49 - 7.67i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 9.10T + 67T^{2} \)
71 \( 1 + (-0.314 - 0.969i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-6.18 - 4.49i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (0.00169 - 0.00520i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-0.875 - 2.69i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 10.5T + 89T^{2} \)
97 \( 1 + (-3.75 + 11.5i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.25592596956882409151166963704, −9.835839403455824345025506332345, −8.433524348863847559461030074370, −7.63707419410544194451184953321, −6.24175856211092035205614867727, −5.49796263842539743168488818969, −4.08381435842978433543738982650, −3.26108661952521081921307846148, −2.48328611748240329058393567715, −0.845949221816514048556538486655, 1.87508807614106118609537422517, 3.68534534818234402429216083500, 4.96355145195766343658001987072, 5.23170203546356308304628403344, 6.66013748718090406370644558227, 6.96790057656850526658590862135, 8.051984267598912353262687866733, 9.012368736467929658226471216249, 9.528827317063333936164417258016, 10.84123310827589851215731335932

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