Properties

Label 2-1694-1.1-c1-0-39
Degree $2$
Conductor $1694$
Sign $-1$
Analytic cond. $13.5266$
Root an. cond. $3.67786$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2·3-s + 4-s − 2·6-s − 7-s + 8-s + 9-s − 2·12-s + 4·13-s − 14-s + 16-s − 6·17-s + 18-s − 2·19-s + 2·21-s − 2·24-s − 5·25-s + 4·26-s + 4·27-s − 28-s + 6·29-s − 4·31-s + 32-s − 6·34-s + 36-s + 2·37-s − 2·38-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.577·12-s + 1.10·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.458·19-s + 0.436·21-s − 0.408·24-s − 25-s + 0.784·26-s + 0.769·27-s − 0.188·28-s + 1.11·29-s − 0.718·31-s + 0.176·32-s − 1.02·34-s + 1/6·36-s + 0.328·37-s − 0.324·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1694 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1694 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1694\)    =    \(2 \cdot 7 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(13.5266\)
Root analytic conductor: \(3.67786\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1694,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−8.888226006934558931566258496787, −8.145906800879012290031802042193, −6.82397964191468499058519060303, −6.42455331216463290634831658204, −5.76321350617402115136418955935, −4.85990751466979898816213085515, −4.09052365620160024762211243831, −3.03039761753313177141617611476, −1.67120553980965395706477292535, 0, 1.67120553980965395706477292535, 3.03039761753313177141617611476, 4.09052365620160024762211243831, 4.85990751466979898816213085515, 5.76321350617402115136418955935, 6.42455331216463290634831658204, 6.82397964191468499058519060303, 8.145906800879012290031802042193, 8.888226006934558931566258496787

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