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Properties

Label 2-1617-77.76-c1-0-3
Degree $2$
Conductor $1617$
Sign $-0.839 - 0.543i$
Analytic cond. $12.9118$
Root an. cond. $3.59330$
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  − 0.802i·2-s + i·3-s + 1.35·4-s − 0.581i·5-s + 0.802·6-s − 2.69i·8-s − 9-s − 0.466·10-s + (−2.78 + 1.80i)11-s + 1.35i·12-s − 2.62·13-s + 0.581·15-s + 0.553·16-s − 7.47·17-s + 0.802i·18-s − 6.46·19-s + ⋯
L(s)  = 1  − 0.567i·2-s + 0.577i·3-s + 0.678·4-s − 0.259i·5-s + 0.327·6-s − 0.951i·8-s − 0.333·9-s − 0.147·10-s + (−0.839 + 0.544i)11-s + 0.391i·12-s − 0.728·13-s + 0.150·15-s + 0.138·16-s − 1.81·17-s + 0.189i·18-s − 1.48·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.839 - 0.543i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $-0.839 - 0.543i$
Analytic conductor: \(12.9118\)
Root analytic conductor: \(3.59330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1617} (538, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1617,\ (\ :1/2),\ -0.839 - 0.543i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1876472782\)
\(L(\frac12)\) \(\approx\) \(0.1876472782\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
7 \( 1 \)
11 \( 1 + (2.78 - 1.80i)T \)
good2 \( 1 + 0.802iT - 2T^{2} \)
5 \( 1 + 0.581iT - 5T^{2} \)
13 \( 1 + 2.62T + 13T^{2} \)
17 \( 1 + 7.47T + 17T^{2} \)
19 \( 1 + 6.46T + 19T^{2} \)
23 \( 1 + 7.03T + 23T^{2} \)
29 \( 1 - 7.12iT - 29T^{2} \)
31 \( 1 - 8.26iT - 31T^{2} \)
37 \( 1 + 4.36T + 37T^{2} \)
41 \( 1 + 3.16T + 41T^{2} \)
43 \( 1 + 11.0iT - 43T^{2} \)
47 \( 1 - 1.37iT - 47T^{2} \)
53 \( 1 - 9.64T + 53T^{2} \)
59 \( 1 - 3.68iT - 59T^{2} \)
61 \( 1 - 14.4T + 61T^{2} \)
67 \( 1 - 9.06T + 67T^{2} \)
71 \( 1 + 6.32T + 71T^{2} \)
73 \( 1 + 4.73T + 73T^{2} \)
79 \( 1 - 5.63iT - 79T^{2} \)
83 \( 1 - 3.74T + 83T^{2} \)
89 \( 1 + 7.87iT - 89T^{2} \)
97 \( 1 - 9.05iT - 97T^{2} \)
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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

−10.10656269353990195478866956120, −8.887287953790509657386185984581, −8.436327453850096100448025265975, −7.06922473512064918052485261139, −6.73902692164574725513296713543, −5.45719436260574777262282824512, −4.63107358251633404292076947188, −3.79427691285426266865723826047, −2.56495783952805507752102732186, −1.95191732367568395289023456791, 0.06005653589479474823193767815, 2.28525416464917984587050267817, 2.41485410943780491057714582450, 4.07819439606133003455958552272, 5.15837641713074813194707262876, 6.22870429048158290235233277545, 6.50019494383827791655826423350, 7.44553493896778497979815634600, 8.144078470985465883319064045470, 8.673524879498369326286210996050

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