Properties

Label 2-2254-1.1-c3-0-92
Degree $2$
Conductor $2254$
Sign $-1$
Analytic cond. $132.990$
Root an. cond. $11.5321$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 7.17·3-s + 4·4-s − 18.7·5-s − 14.3·6-s + 8·8-s + 24.5·9-s − 37.4·10-s − 36.8·11-s − 28.7·12-s − 22.7·13-s + 134.·15-s + 16·16-s − 66.6·17-s + 49.0·18-s + 106.·19-s − 74.8·20-s − 73.7·22-s + 23·23-s − 57.4·24-s + 224.·25-s − 45.5·26-s + 17.8·27-s − 199.·29-s + 268.·30-s + 262.·31-s + 32·32-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.38·3-s + 0.5·4-s − 1.67·5-s − 0.976·6-s + 0.353·8-s + 0.907·9-s − 1.18·10-s − 1.01·11-s − 0.690·12-s − 0.486·13-s + 2.31·15-s + 0.250·16-s − 0.951·17-s + 0.641·18-s + 1.28·19-s − 0.836·20-s − 0.715·22-s + 0.208·23-s − 0.488·24-s + 1.79·25-s − 0.343·26-s + 0.127·27-s − 1.27·29-s + 1.63·30-s + 1.51·31-s + 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2254\)    =    \(2 \cdot 7^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(132.990\)
Root analytic conductor: \(11.5321\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2254,\ (\ :3/2),\ -1)\)

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 \( 1 - 2T \)
7 \( 1 \)
23 \( 1 - 23T \)
good3 \( 1 + 7.17T + 27T^{2} \)
5 \( 1 + 18.7T + 125T^{2} \)
11 \( 1 + 36.8T + 1.33e3T^{2} \)
13 \( 1 + 22.7T + 2.19e3T^{2} \)
17 \( 1 + 66.6T + 4.91e3T^{2} \)
19 \( 1 - 106.T + 6.85e3T^{2} \)
29 \( 1 + 199.T + 2.43e4T^{2} \)
31 \( 1 - 262.T + 2.97e4T^{2} \)
37 \( 1 - 202.T + 5.06e4T^{2} \)
41 \( 1 + 44.4T + 6.89e4T^{2} \)
43 \( 1 + 18.2T + 7.95e4T^{2} \)
47 \( 1 - 306.T + 1.03e5T^{2} \)
53 \( 1 - 300.T + 1.48e5T^{2} \)
59 \( 1 + 255.T + 2.05e5T^{2} \)
61 \( 1 + 598.T + 2.26e5T^{2} \)
67 \( 1 + 636.T + 3.00e5T^{2} \)
71 \( 1 - 320.T + 3.57e5T^{2} \)
73 \( 1 - 24.5T + 3.89e5T^{2} \)
79 \( 1 - 1.12e3T + 4.93e5T^{2} \)
83 \( 1 - 767.T + 5.71e5T^{2} \)
89 \( 1 - 807.T + 7.04e5T^{2} \)
97 \( 1 + 1.17e3T + 9.12e5T^{2} \)
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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

−7.81668494591790018903695330043, −7.50722505410815247527426742747, −6.65409063275538606004758090844, −5.79175468821126784880144600066, −4.92195948375850964002006743786, −4.55500603692004924480532023663, −3.54063398001694768403749582154, −2.57879205759713552607542495647, −0.846043750499627901839896537967, 0, 0.846043750499627901839896537967, 2.57879205759713552607542495647, 3.54063398001694768403749582154, 4.55500603692004924480532023663, 4.92195948375850964002006743786, 5.79175468821126784880144600066, 6.65409063275538606004758090844, 7.50722505410815247527426742747, 7.81668494591790018903695330043

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