Properties

Label 2-1617-1.1-c1-0-0
Degree $2$
Conductor $1617$
Sign $1$
Analytic cond. $12.9118$
Root an. cond. $3.59330$
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  − 1.93·2-s − 3-s + 1.74·4-s − 4.18·5-s + 1.93·6-s + 0.491·8-s + 9-s + 8.10·10-s − 11-s − 1.74·12-s + 3.17·13-s + 4.18·15-s − 4.44·16-s − 6.85·17-s − 1.93·18-s + 0.318·19-s − 7.31·20-s + 1.93·22-s − 1.87·23-s − 0.491·24-s + 12.5·25-s − 6.14·26-s − 27-s − 3.17·29-s − 8.10·30-s − 9.23·31-s + 7.61·32-s + ⋯
L(s)  = 1  − 1.36·2-s − 0.577·3-s + 0.872·4-s − 1.87·5-s + 0.790·6-s + 0.173·8-s + 0.333·9-s + 2.56·10-s − 0.301·11-s − 0.503·12-s + 0.880·13-s + 1.08·15-s − 1.11·16-s − 1.66·17-s − 0.456·18-s + 0.0731·19-s − 1.63·20-s + 0.412·22-s − 0.390·23-s − 0.100·24-s + 2.51·25-s − 1.20·26-s − 0.192·27-s − 0.589·29-s − 1.48·30-s − 1.65·31-s + 1.34·32-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(0.1583913633\)
\(L(\frac12)\) \(\approx\) \(0.1583913633\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 \)
11 \( 1 + T \)
good2 \( 1 + 1.93T + 2T^{2} \)
5 \( 1 + 4.18T + 5T^{2} \)
13 \( 1 - 3.17T + 13T^{2} \)
17 \( 1 + 6.85T + 17T^{2} \)
19 \( 1 - 0.318T + 19T^{2} \)
23 \( 1 + 1.87T + 23T^{2} \)
29 \( 1 + 3.17T + 29T^{2} \)
31 \( 1 + 9.23T + 31T^{2} \)
37 \( 1 + 7.55T + 37T^{2} \)
41 \( 1 + 9.36T + 41T^{2} \)
43 \( 1 + 10.8T + 43T^{2} \)
47 \( 1 - 8.06T + 47T^{2} \)
53 \( 1 - 0.508T + 53T^{2} \)
59 \( 1 - 7.04T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 2.66T + 67T^{2} \)
71 \( 1 + 5.01T + 71T^{2} \)
73 \( 1 - 4.82T + 73T^{2} \)
79 \( 1 - 5.01T + 79T^{2} \)
83 \( 1 + 3.52T + 83T^{2} \)
89 \( 1 - 1.74T + 89T^{2} \)
97 \( 1 - 12.2T + 97T^{2} \)
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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

−9.099866118040211515142971353235, −8.598634653417493343199553323455, −7.983271024894237096239245526485, −7.12184108580959852025660763627, −6.72150782913140996364466348896, −5.23220885059817861860237193754, −4.26951023394719819426710570509, −3.53107319398695235484402346659, −1.81651711820254547121141008796, −0.34547448699083159288643690383, 0.34547448699083159288643690383, 1.81651711820254547121141008796, 3.53107319398695235484402346659, 4.26951023394719819426710570509, 5.23220885059817861860237193754, 6.72150782913140996364466348896, 7.12184108580959852025660763627, 7.983271024894237096239245526485, 8.598634653417493343199553323455, 9.099866118040211515142971353235

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