Properties

Label 2-671-1.1-c1-0-45
Degree $2$
Conductor $671$
Sign $1$
Analytic cond. $5.35796$
Root an. cond. $2.31472$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.61·2-s + 2.06·3-s + 0.609·4-s + 2.50·5-s + 3.34·6-s + 4.06·7-s − 2.24·8-s + 1.28·9-s + 4.03·10-s − 11-s + 1.26·12-s − 6.62·13-s + 6.56·14-s + 5.17·15-s − 4.84·16-s − 0.430·17-s + 2.06·18-s − 6.80·19-s + 1.52·20-s + 8.40·21-s − 1.61·22-s + 3.37·23-s − 4.64·24-s + 1.25·25-s − 10.7·26-s − 3.55·27-s + 2.47·28-s + ⋯
L(s)  = 1  + 1.14·2-s + 1.19·3-s + 0.304·4-s + 1.11·5-s + 1.36·6-s + 1.53·7-s − 0.794·8-s + 0.426·9-s + 1.27·10-s − 0.301·11-s + 0.363·12-s − 1.83·13-s + 1.75·14-s + 1.33·15-s − 1.21·16-s − 0.104·17-s + 0.487·18-s − 1.56·19-s + 0.340·20-s + 1.83·21-s − 0.344·22-s + 0.704·23-s − 0.948·24-s + 0.250·25-s − 2.10·26-s − 0.684·27-s + 0.467·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(671\)    =    \(11 \cdot 61\)
Sign: $1$
Analytic conductor: \(5.35796\)
Root analytic conductor: \(2.31472\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 671,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.129603232\)
\(L(\frac12)\) \(\approx\) \(4.129603232\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + T \)
61 \( 1 - T \)
good2 \( 1 - 1.61T + 2T^{2} \)
3 \( 1 - 2.06T + 3T^{2} \)
5 \( 1 - 2.50T + 5T^{2} \)
7 \( 1 - 4.06T + 7T^{2} \)
13 \( 1 + 6.62T + 13T^{2} \)
17 \( 1 + 0.430T + 17T^{2} \)
19 \( 1 + 6.80T + 19T^{2} \)
23 \( 1 - 3.37T + 23T^{2} \)
29 \( 1 - 6.96T + 29T^{2} \)
31 \( 1 + 7.86T + 31T^{2} \)
37 \( 1 - 5.73T + 37T^{2} \)
41 \( 1 + 7.27T + 41T^{2} \)
43 \( 1 - 10.2T + 43T^{2} \)
47 \( 1 - 5.00T + 47T^{2} \)
53 \( 1 - 3.76T + 53T^{2} \)
59 \( 1 - 9.01T + 59T^{2} \)
67 \( 1 - 9.04T + 67T^{2} \)
71 \( 1 - 11.6T + 71T^{2} \)
73 \( 1 + 7.40T + 73T^{2} \)
79 \( 1 - 4.79T + 79T^{2} \)
83 \( 1 + 2.01T + 83T^{2} \)
89 \( 1 + 12.0T + 89T^{2} \)
97 \( 1 - 8.82T + 97T^{2} \)
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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

−10.48291269186407148439194620100, −9.490164236299597418696591591861, −8.785021503925953836881837782023, −7.989219036847855077960265566718, −6.92365937792895412501261178029, −5.61209343691832215887640228627, −4.94958014634942631262158516189, −4.11219523409106851419647974160, −2.53884243209767189797863047940, −2.19230094052982981533866968855, 2.19230094052982981533866968855, 2.53884243209767189797863047940, 4.11219523409106851419647974160, 4.94958014634942631262158516189, 5.61209343691832215887640228627, 6.92365937792895412501261178029, 7.989219036847855077960265566718, 8.785021503925953836881837782023, 9.490164236299597418696591591861, 10.48291269186407148439194620100

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