Properties

Label 2-1617-77.76-c1-0-33
Degree $2$
Conductor $1617$
Sign $0.381 - 0.924i$
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  − 0.655i·2-s + i·3-s + 1.57·4-s + 3.40i·5-s + 0.655·6-s − 2.33i·8-s − 9-s + 2.22·10-s + (1.48 + 2.96i)11-s + 1.57i·12-s + 5.44·13-s − 3.40·15-s + 1.60·16-s − 1.70·17-s + 0.655i·18-s + 6.25·19-s + ⋯
L(s)  = 1  − 0.463i·2-s + 0.577i·3-s + 0.785·4-s + 1.52i·5-s + 0.267·6-s − 0.827i·8-s − 0.333·9-s + 0.704·10-s + (0.448 + 0.893i)11-s + 0.453i·12-s + 1.51·13-s − 0.878·15-s + 0.402·16-s − 0.412·17-s + 0.154i·18-s + 1.43·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.381 - 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.381 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $0.381 - 0.924i$
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: \((2,\ 1617,\ (\ :1/2),\ 0.381 - 0.924i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.293390244\)
\(L(\frac12)\) \(\approx\) \(2.293390244\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.48 - 2.96i)T \)
good2 \( 1 + 0.655iT - 2T^{2} \)
5 \( 1 - 3.40iT - 5T^{2} \)
13 \( 1 - 5.44T + 13T^{2} \)
17 \( 1 + 1.70T + 17T^{2} \)
19 \( 1 - 6.25T + 19T^{2} \)
23 \( 1 + 6.18T + 23T^{2} \)
29 \( 1 + 3.61iT - 29T^{2} \)
31 \( 1 - 6.50iT - 31T^{2} \)
37 \( 1 - 0.165T + 37T^{2} \)
41 \( 1 + 1.35T + 41T^{2} \)
43 \( 1 + 3.12iT - 43T^{2} \)
47 \( 1 - 0.800iT - 47T^{2} \)
53 \( 1 - 6.40T + 53T^{2} \)
59 \( 1 - 1.78iT - 59T^{2} \)
61 \( 1 - 3.56T + 61T^{2} \)
67 \( 1 + 8.89T + 67T^{2} \)
71 \( 1 - 3.59T + 71T^{2} \)
73 \( 1 + 10.8T + 73T^{2} \)
79 \( 1 + 4.73iT - 79T^{2} \)
83 \( 1 + 14.2T + 83T^{2} \)
89 \( 1 + 3.82iT - 89T^{2} \)
97 \( 1 - 3.87iT - 97T^{2} \)
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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

−9.968978404927553878613839452436, −8.948041896735703817305131155809, −7.78843050951098903397062153898, −7.03278254479175899784732535678, −6.40878497998994861070978210715, −5.66507272586722577811478141596, −4.12032461526506476777402286230, −3.47491960098324144074366756735, −2.67625672425589544162423155730, −1.58562073120137576182791198913, 0.945448158143654190459566598022, 1.74412239932683414530043736764, 3.20613443833151561854696038670, 4.26857809892196789547145632917, 5.67819297677159342065292577192, 5.77092374378559383686866224952, 6.79947674702716522855227315998, 7.76788142240930395957563344737, 8.421034119927587931463080098408, 8.847752325149991503261072828318

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