Properties

Degree $2$
Conductor $1849$
Sign $-1$
Motivic weight $3$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.80·2-s − 2.56·3-s + 15.0·4-s − 6.34·5-s − 12.3·6-s + 6.17·7-s + 34.0·8-s − 20.4·9-s − 30.4·10-s + 33.6·11-s − 38.6·12-s − 34.3·13-s + 29.6·14-s + 16.2·15-s + 42.7·16-s + 105.·17-s − 98.0·18-s + 14.3·19-s − 95.6·20-s − 15.8·21-s + 161.·22-s − 160.·23-s − 87.2·24-s − 84.7·25-s − 165.·26-s + 121.·27-s + 93.1·28-s + ⋯
L(s)  = 1  + 1.69·2-s − 0.493·3-s + 1.88·4-s − 0.567·5-s − 0.838·6-s + 0.333·7-s + 1.50·8-s − 0.756·9-s − 0.963·10-s + 0.921·11-s − 0.930·12-s − 0.733·13-s + 0.566·14-s + 0.280·15-s + 0.668·16-s + 1.51·17-s − 1.28·18-s + 0.172·19-s − 1.06·20-s − 0.164·21-s + 1.56·22-s − 1.45·23-s − 0.742·24-s − 0.678·25-s − 1.24·26-s + 0.867·27-s + 0.628·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1849\)    =    \(43^{2}\)
Sign: $-1$
Motivic weight: \(3\)
Character: $\chi_{1849} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1849,\ (\ :3/2),\ -1)\)

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)$
bad43 \( 1 \)
good2 \( 1 - 4.80T + 8T^{2} \)
3 \( 1 + 2.56T + 27T^{2} \)
5 \( 1 + 6.34T + 125T^{2} \)
7 \( 1 - 6.17T + 343T^{2} \)
11 \( 1 - 33.6T + 1.33e3T^{2} \)
13 \( 1 + 34.3T + 2.19e3T^{2} \)
17 \( 1 - 105.T + 4.91e3T^{2} \)
19 \( 1 - 14.3T + 6.85e3T^{2} \)
23 \( 1 + 160.T + 1.21e4T^{2} \)
29 \( 1 + 42.2T + 2.43e4T^{2} \)
31 \( 1 - 0.100T + 2.97e4T^{2} \)
37 \( 1 + 369.T + 5.06e4T^{2} \)
41 \( 1 - 450.T + 6.89e4T^{2} \)
47 \( 1 + 288.T + 1.03e5T^{2} \)
53 \( 1 - 439.T + 1.48e5T^{2} \)
59 \( 1 + 16.1T + 2.05e5T^{2} \)
61 \( 1 + 764.T + 2.26e5T^{2} \)
67 \( 1 + 415.T + 3.00e5T^{2} \)
71 \( 1 - 998.T + 3.57e5T^{2} \)
73 \( 1 - 160.T + 3.89e5T^{2} \)
79 \( 1 + 1.31e3T + 4.93e5T^{2} \)
83 \( 1 + 374.T + 5.71e5T^{2} \)
89 \( 1 + 706.T + 7.04e5T^{2} \)
97 \( 1 + 1.26e3T + 9.12e5T^{2} \)
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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

−8.170167234183166987528816782911, −7.48783557480533943524091033218, −6.57847460258063639659167733715, −5.76165690963034763008649963967, −5.32192459343914367509026225882, −4.32436948223746907218004468547, −3.68537485438107895707500119824, −2.81328408568505850894365144300, −1.59755607500392249822990982821, 0, 1.59755607500392249822990982821, 2.81328408568505850894365144300, 3.68537485438107895707500119824, 4.32436948223746907218004468547, 5.32192459343914367509026225882, 5.76165690963034763008649963967, 6.57847460258063639659167733715, 7.48783557480533943524091033218, 8.170167234183166987528816782911

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