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Properties

Label 2-693-11.5-c1-0-17
Degree $2$
Conductor $693$
Sign $0.762 - 0.647i$
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  + (0.435 + 1.34i)2-s + (0.0112 − 0.00820i)4-s + (0.565 − 1.74i)5-s + (−0.809 + 0.587i)7-s + (2.29 + 1.66i)8-s + 2.58·10-s + (2.26 + 2.42i)11-s + (−1.43 − 4.41i)13-s + (−1.14 − 0.828i)14-s + (−1.22 + 3.77i)16-s + (1.69 − 5.20i)17-s + (4.69 + 3.40i)19-s + (−0.00790 − 0.0243i)20-s + (−2.26 + 4.08i)22-s + 0.719·23-s + ⋯
L(s)  = 1  + (0.307 + 0.947i)2-s + (0.00564 − 0.00410i)4-s + (0.253 − 0.778i)5-s + (−0.305 + 0.222i)7-s + (0.811 + 0.589i)8-s + 0.816·10-s + (0.681 + 0.731i)11-s + (−0.398 − 1.22i)13-s + (−0.304 − 0.221i)14-s + (−0.306 + 0.944i)16-s + (0.409 − 1.26i)17-s + (1.07 + 0.782i)19-s + (−0.00176 − 0.00543i)20-s + (−0.483 + 0.871i)22-s + 0.150·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 - 0.647i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.762 - 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.762 - 0.647i$
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: \((2,\ 693,\ (\ :1/2),\ 0.762 - 0.647i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.01566 + 0.740316i\)
\(L(\frac12)\) \(\approx\) \(2.01566 + 0.740316i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.26 - 2.42i)T \)
good2 \( 1 + (-0.435 - 1.34i)T + (-1.61 + 1.17i)T^{2} \)
5 \( 1 + (-0.565 + 1.74i)T + (-4.04 - 2.93i)T^{2} \)
13 \( 1 + (1.43 + 4.41i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.69 + 5.20i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-4.69 - 3.40i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 0.719T + 23T^{2} \)
29 \( 1 + (0.948 - 0.689i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.404 + 1.24i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (1.69 - 1.23i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (0.741 + 0.538i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 8.02T + 43T^{2} \)
47 \( 1 + (-4.83 - 3.51i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (3.13 + 9.64i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (6.21 - 4.51i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (1.93 - 5.96i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 15.4T + 67T^{2} \)
71 \( 1 + (4.29 - 13.2i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (4.86 - 3.53i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (4.83 + 14.8i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-1.35 + 4.16i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 15.3T + 89T^{2} \)
97 \( 1 + (-0.745 - 2.29i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.34076681297083695325913964141, −9.619641689240778675324177223515, −8.775258172111641580309681090018, −7.58448080023875142484593378924, −7.19898110066949570934713097604, −5.88528870624202699114880081771, −5.36539559413404845921410380977, −4.49651293507335929272517955719, −2.92930317891507498466528272696, −1.32111391590392936561389170428, 1.43579519062344877744091210025, 2.71559041963088538153763844214, 3.55582728439480502145143666473, 4.47192191607680958824450173619, 6.00480572253872320709542652752, 6.82966802668182625247296618277, 7.54621507643164590481031789180, 8.949062255861082461428937796914, 9.713546385684601758320777982538, 10.67866725524640585739657286948

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