Properties

Label 2-1859-1.1-c3-0-286
Degree $2$
Conductor $1859$
Sign $-1$
Analytic cond. $109.684$
Root an. cond. $10.4730$
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  − 3.71·2-s + 7.61·3-s + 5.82·4-s − 15.9·5-s − 28.3·6-s + 21.9·7-s + 8.07·8-s + 31.0·9-s + 59.4·10-s + 11·11-s + 44.3·12-s − 81.6·14-s − 121.·15-s − 76.6·16-s − 94.2·17-s − 115.·18-s − 13.8·19-s − 93.1·20-s + 167.·21-s − 40.9·22-s + 77.7·23-s + 61.5·24-s + 130.·25-s + 30.6·27-s + 127.·28-s + 295.·29-s + 452.·30-s + ⋯
L(s)  = 1  − 1.31·2-s + 1.46·3-s + 0.728·4-s − 1.42·5-s − 1.92·6-s + 1.18·7-s + 0.356·8-s + 1.14·9-s + 1.87·10-s + 0.301·11-s + 1.06·12-s − 1.55·14-s − 2.09·15-s − 1.19·16-s − 1.34·17-s − 1.51·18-s − 0.167·19-s − 1.04·20-s + 1.73·21-s − 0.396·22-s + 0.704·23-s + 0.523·24-s + 1.04·25-s + 0.218·27-s + 0.863·28-s + 1.89·29-s + 2.75·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1859\)    =    \(11 \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(109.684\)
Root analytic conductor: \(10.4730\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1859,\ (\ :3/2),\ -1)\)

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)$
bad11 \( 1 - 11T \)
13 \( 1 \)
good2 \( 1 + 3.71T + 8T^{2} \)
3 \( 1 - 7.61T + 27T^{2} \)
5 \( 1 + 15.9T + 125T^{2} \)
7 \( 1 - 21.9T + 343T^{2} \)
17 \( 1 + 94.2T + 4.91e3T^{2} \)
19 \( 1 + 13.8T + 6.85e3T^{2} \)
23 \( 1 - 77.7T + 1.21e4T^{2} \)
29 \( 1 - 295.T + 2.43e4T^{2} \)
31 \( 1 + 136.T + 2.97e4T^{2} \)
37 \( 1 - 145.T + 5.06e4T^{2} \)
41 \( 1 + 324.T + 6.89e4T^{2} \)
43 \( 1 + 287.T + 7.95e4T^{2} \)
47 \( 1 + 413.T + 1.03e5T^{2} \)
53 \( 1 - 522.T + 1.48e5T^{2} \)
59 \( 1 - 215.T + 2.05e5T^{2} \)
61 \( 1 + 862.T + 2.26e5T^{2} \)
67 \( 1 - 281.T + 3.00e5T^{2} \)
71 \( 1 - 768.T + 3.57e5T^{2} \)
73 \( 1 + 767.T + 3.89e5T^{2} \)
79 \( 1 + 641.T + 4.93e5T^{2} \)
83 \( 1 - 193.T + 5.71e5T^{2} \)
89 \( 1 + 758.T + 7.04e5T^{2} \)
97 \( 1 + 994.T + 9.12e5T^{2} \)
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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

−8.354217875650718277288745879391, −8.231565611720714848708677188749, −7.32268129927687243637270474175, −6.79791192590110857412951869125, −4.76586324250779600846861474934, −4.30314886698882745434865681892, −3.25448833741134825912851955746, −2.17151005777329305267960600670, −1.26199295167186834771247081838, 0, 1.26199295167186834771247081838, 2.17151005777329305267960600670, 3.25448833741134825912851955746, 4.30314886698882745434865681892, 4.76586324250779600846861474934, 6.79791192590110857412951869125, 7.32268129927687243637270474175, 8.231565611720714848708677188749, 8.354217875650718277288745879391

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