Properties

Label 2-1682-1.1-c3-0-134
Degree $2$
Conductor $1682$
Sign $-1$
Analytic cond. $99.2412$
Root an. cond. $9.96198$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 7·3-s + 4·4-s + 5·5-s − 14·6-s − 2·7-s + 8·8-s + 22·9-s + 10·10-s − 37·11-s − 28·12-s + 27·13-s − 4·14-s − 35·15-s + 16·16-s − 24·17-s + 44·18-s + 88·19-s + 20·20-s + 14·21-s − 74·22-s − 28·23-s − 56·24-s − 100·25-s + 54·26-s + 35·27-s − 8·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.34·3-s + 1/2·4-s + 0.447·5-s − 0.952·6-s − 0.107·7-s + 0.353·8-s + 0.814·9-s + 0.316·10-s − 1.01·11-s − 0.673·12-s + 0.576·13-s − 0.0763·14-s − 0.602·15-s + 1/4·16-s − 0.342·17-s + 0.576·18-s + 1.06·19-s + 0.223·20-s + 0.145·21-s − 0.717·22-s − 0.253·23-s − 0.476·24-s − 4/5·25-s + 0.407·26-s + 0.249·27-s − 0.0539·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1682\)    =    \(2 \cdot 29^{2}\)
Sign: $-1$
Analytic conductor: \(99.2412\)
Root analytic conductor: \(9.96198\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1682,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p T \)
29 \( 1 \)
good3 \( 1 + 7 T + p^{3} T^{2} \)
5 \( 1 - p T + p^{3} T^{2} \)
7 \( 1 + 2 T + p^{3} T^{2} \)
11 \( 1 + 37 T + p^{3} T^{2} \)
13 \( 1 - 27 T + p^{3} T^{2} \)
17 \( 1 + 24 T + p^{3} T^{2} \)
19 \( 1 - 88 T + p^{3} T^{2} \)
23 \( 1 + 28 T + p^{3} T^{2} \)
31 \( 1 - 143 T + p^{3} T^{2} \)
37 \( 1 - 360 T + p^{3} T^{2} \)
41 \( 1 + 386 T + p^{3} T^{2} \)
43 \( 1 + 381 T + p^{3} T^{2} \)
47 \( 1 - 103 T + p^{3} T^{2} \)
53 \( 1 + 431 T + p^{3} T^{2} \)
59 \( 1 - 288 T + p^{3} T^{2} \)
61 \( 1 - 840 T + p^{3} T^{2} \)
67 \( 1 + 180 T + p^{3} T^{2} \)
71 \( 1 - 706 T + p^{3} T^{2} \)
73 \( 1 + 716 T + p^{3} T^{2} \)
79 \( 1 + 931 T + p^{3} T^{2} \)
83 \( 1 - 1188 T + p^{3} T^{2} \)
89 \( 1 - 642 T + p^{3} T^{2} \)
97 \( 1 + 486 T + p^{3} T^{2} \)
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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

−8.441617123513745835688644171747, −7.59965450257279681288745896696, −6.55439064006180327721043259127, −6.08857962856197489945599053109, −5.27703245772853535763835974397, −4.81400806237131701710144433923, −3.60038134971572399454830269143, −2.50844294831584910969985138310, −1.25004343657418131865356580716, 0, 1.25004343657418131865356580716, 2.50844294831584910969985138310, 3.60038134971572399454830269143, 4.81400806237131701710144433923, 5.27703245772853535763835974397, 6.08857962856197489945599053109, 6.55439064006180327721043259127, 7.59965450257279681288745896696, 8.441617123513745835688644171747

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