Properties

Label 2-1710-5.4-c1-0-5
Degree $2$
Conductor $1710$
Sign $0.783 - 0.621i$
Analytic cond. $13.6544$
Root an. cond. $3.69518$
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  i·2-s − 4-s + (−1.38 − 1.75i)5-s + 0.636i·7-s + i·8-s + (−1.75 + 1.38i)10-s − 3.50·11-s + 0.141i·13-s + 0.636·14-s + 16-s − 2.14i·17-s − 19-s + (1.38 + 1.75i)20-s + 3.50i·22-s + 4.91i·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (−0.621 − 0.783i)5-s + 0.240i·7-s + 0.353i·8-s + (−0.554 + 0.439i)10-s − 1.05·11-s + 0.0391i·13-s + 0.170·14-s + 0.250·16-s − 0.519i·17-s − 0.229·19-s + (0.310 + 0.391i)20-s + 0.747i·22-s + 1.02i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.783 - 0.621i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.783 - 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1710\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $0.783 - 0.621i$
Analytic conductor: \(13.6544\)
Root analytic conductor: \(3.69518\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1710} (1369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1710,\ (\ :1/2),\ 0.783 - 0.621i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6788910342\)
\(L(\frac12)\) \(\approx\) \(0.6788910342\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
5 \( 1 + (1.38 + 1.75i)T \)
19 \( 1 + T \)
good7 \( 1 - 0.636iT - 7T^{2} \)
11 \( 1 + 3.50T + 11T^{2} \)
13 \( 1 - 0.141iT - 13T^{2} \)
17 \( 1 + 2.14iT - 17T^{2} \)
23 \( 1 - 4.91iT - 23T^{2} \)
29 \( 1 - 7.15T + 29T^{2} \)
31 \( 1 + 7.78T + 31T^{2} \)
37 \( 1 - 3.27iT - 37T^{2} \)
41 \( 1 - 4.23T + 41T^{2} \)
43 \( 1 + 2.49iT - 43T^{2} \)
47 \( 1 - 10.2iT - 47T^{2} \)
53 \( 1 - 8.14iT - 53T^{2} \)
59 \( 1 + 5.64T + 59T^{2} \)
61 \( 1 + 6.49T + 61T^{2} \)
67 \( 1 - 8.37iT - 67T^{2} \)
71 \( 1 - 8.95T + 71T^{2} \)
73 \( 1 - 3.69iT - 73T^{2} \)
79 \( 1 - 4.17T + 79T^{2} \)
83 \( 1 + 9.00iT - 83T^{2} \)
89 \( 1 + 6.77T + 89T^{2} \)
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

−9.322244088876908594902047505534, −8.809900800181595804446707499643, −7.88784345992867184527334080945, −7.36811582992396403499578502238, −5.93281555973512734940679126665, −5.12066751269161422079203444382, −4.44409850700301929408574125473, −3.40827036480113921532900802360, −2.44976006038993097396497868064, −1.11747570046908030427770617744, 0.29177037184855514264854367761, 2.32088951410355913559779022034, 3.40740267045391422776335282650, 4.29972417363840354773689967441, 5.21028534647412252353111496706, 6.20525382471038788800965796905, 6.89065747210411967377976573852, 7.67241201912565368196418745885, 8.215248981284183430401736596969, 9.031576973199655752478889772822

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