Properties

Label 2-1694-1.1-c3-0-43
Degree $2$
Conductor $1694$
Sign $-1$
Analytic cond. $99.9492$
Root an. cond. $9.99746$
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 − 10·3-s + 4·4-s − 14·5-s + 20·6-s − 7·7-s − 8·8-s + 73·9-s + 28·10-s − 40·12-s + 16·13-s + 14·14-s + 140·15-s + 16·16-s − 108·17-s − 146·18-s − 116·19-s − 56·20-s + 70·21-s + 68·23-s + 80·24-s + 71·25-s − 32·26-s − 460·27-s − 28·28-s − 122·29-s − 280·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.92·3-s + 1/2·4-s − 1.25·5-s + 1.36·6-s − 0.377·7-s − 0.353·8-s + 2.70·9-s + 0.885·10-s − 0.962·12-s + 0.341·13-s + 0.267·14-s + 2.40·15-s + 1/4·16-s − 1.54·17-s − 1.91·18-s − 1.40·19-s − 0.626·20-s + 0.727·21-s + 0.616·23-s + 0.680·24-s + 0.567·25-s − 0.241·26-s − 3.27·27-s − 0.188·28-s − 0.781·29-s − 1.70·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1694 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1694 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1694\)    =    \(2 \cdot 7 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(99.9492\)
Root analytic conductor: \(9.99746\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1694} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1694,\ (\ :3/2),\ -1)\)

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 \( 1 + p T \)
7 \( 1 + p T \)
11 \( 1 \)
good3 \( 1 + 10 T + p^{3} T^{2} \)
5 \( 1 + 14 T + p^{3} T^{2} \)
13 \( 1 - 16 T + p^{3} T^{2} \)
17 \( 1 + 108 T + p^{3} T^{2} \)
19 \( 1 + 116 T + p^{3} T^{2} \)
23 \( 1 - 68 T + p^{3} T^{2} \)
29 \( 1 + 122 T + p^{3} T^{2} \)
31 \( 1 + 262 T + p^{3} T^{2} \)
37 \( 1 - 130 T + p^{3} T^{2} \)
41 \( 1 + 204 T + p^{3} T^{2} \)
43 \( 1 - 396 T + p^{3} T^{2} \)
47 \( 1 - 166 T + p^{3} T^{2} \)
53 \( 1 - 442 T + p^{3} T^{2} \)
59 \( 1 - 702 T + p^{3} T^{2} \)
61 \( 1 + 196 T + p^{3} T^{2} \)
67 \( 1 + 416 T + p^{3} T^{2} \)
71 \( 1 - 492 T + p^{3} T^{2} \)
73 \( 1 + 408 T + p^{3} T^{2} \)
79 \( 1 + 600 T + p^{3} T^{2} \)
83 \( 1 - 1212 T + p^{3} T^{2} \)
89 \( 1 - 1146 T + p^{3} T^{2} \)
97 \( 1 + 482 T + p^{3} T^{2} \)
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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

−8.653075588159087106207614458049, −7.52139679557432652668301801388, −6.97037971529508645208632003614, −6.32422363311748966097827519475, −5.48762927579596449528563891079, −4.38363521269410752090390435368, −3.84601425042728243517306783120, −2.03845730714803898142598865866, −0.65903119127355310029922227797, 0, 0.65903119127355310029922227797, 2.03845730714803898142598865866, 3.84601425042728243517306783120, 4.38363521269410752090390435368, 5.48762927579596449528563891079, 6.32422363311748966097827519475, 6.97037971529508645208632003614, 7.52139679557432652668301801388, 8.653075588159087106207614458049

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