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Properties

Label 2-1617-77.76-c1-0-46
Degree $2$
Conductor $1617$
Sign $0.251 + 0.967i$
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  − 1.01i·2-s − i·3-s + 0.969·4-s + 1.37i·5-s − 1.01·6-s − 3.01i·8-s − 9-s + 1.39·10-s + (1.32 + 3.03i)11-s − 0.969i·12-s − 0.625·13-s + 1.37·15-s − 1.11·16-s − 1.13·17-s + 1.01i·18-s + 3.04·19-s + ⋯
L(s)  = 1  − 0.717i·2-s − 0.577i·3-s + 0.484·4-s + 0.612i·5-s − 0.414·6-s − 1.06i·8-s − 0.333·9-s + 0.439·10-s + (0.400 + 0.916i)11-s − 0.279i·12-s − 0.173·13-s + 0.353·15-s − 0.279·16-s − 0.275·17-s + 0.239i·18-s + 0.698·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.251 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.251 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $0.251 + 0.967i$
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: \((2,\ 1617,\ (\ :1/2),\ 0.251 + 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.154507624\)
\(L(\frac12)\) \(\approx\) \(2.154507624\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.32 - 3.03i)T \)
good2 \( 1 + 1.01iT - 2T^{2} \)
5 \( 1 - 1.37iT - 5T^{2} \)
13 \( 1 + 0.625T + 13T^{2} \)
17 \( 1 + 1.13T + 17T^{2} \)
19 \( 1 - 3.04T + 19T^{2} \)
23 \( 1 - 7.31T + 23T^{2} \)
29 \( 1 + 5.55iT - 29T^{2} \)
31 \( 1 + 2.26iT - 31T^{2} \)
37 \( 1 - 4.09T + 37T^{2} \)
41 \( 1 + 7.17T + 41T^{2} \)
43 \( 1 + 3.72iT - 43T^{2} \)
47 \( 1 - 3.00iT - 47T^{2} \)
53 \( 1 - 0.473T + 53T^{2} \)
59 \( 1 - 3.07iT - 59T^{2} \)
61 \( 1 - 13.7T + 61T^{2} \)
67 \( 1 - 4.78T + 67T^{2} \)
71 \( 1 - 3.67T + 71T^{2} \)
73 \( 1 - 13.2T + 73T^{2} \)
79 \( 1 - 2.26iT - 79T^{2} \)
83 \( 1 + 8.93T + 83T^{2} \)
89 \( 1 - 7.47iT - 89T^{2} \)
97 \( 1 + 14.5iT - 97T^{2} \)
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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

−9.545517208689753198254191035193, −8.430148820366436421653762672692, −7.28001160287284463156190547347, −7.01536016330530252590452523140, −6.23889141545778252152282002115, −5.05834444265085223423216970962, −3.88255807309257620908172983922, −2.88111590170917527446966271935, −2.17703311880587806967799323392, −1.01720031236781686339629626195, 1.16136217991196892824566491260, 2.72890229730158629729864039769, 3.62072732763225284522438576955, 5.06419311882212017694015065185, 5.24341617012905665878463669867, 6.44497234594519280612532571297, 7.00610493274811452655016933342, 8.072427587473469351685662296573, 8.724010762757889826830588007000, 9.278539192842168048070232495866

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