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  + 1.29·2-s − 3.36·3-s − 6.32·4-s + 1.78·5-s − 4.34·6-s − 27.5·7-s − 18.5·8-s − 15.7·9-s + 2.30·10-s + 9.45·11-s + 21.2·12-s + 38.4·13-s − 35.6·14-s − 5.98·15-s + 26.6·16-s + 54.0·17-s − 20.3·18-s + 37.3·19-s − 11.2·20-s + 92.6·21-s + 12.2·22-s − 100.·23-s + 62.2·24-s − 121.·25-s + 49.7·26-s + 143.·27-s + 174.·28-s + ⋯
L(s)  = 1  + 0.457·2-s − 0.646·3-s − 0.790·4-s + 0.159·5-s − 0.295·6-s − 1.48·7-s − 0.819·8-s − 0.581·9-s + 0.0729·10-s + 0.259·11-s + 0.511·12-s + 0.820·13-s − 0.681·14-s − 0.103·15-s + 0.415·16-s + 0.771·17-s − 0.266·18-s + 0.450·19-s − 0.125·20-s + 0.962·21-s + 0.118·22-s − 0.915·23-s + 0.529·24-s − 0.974·25-s + 0.375·26-s + 1.02·27-s + 1.17·28-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 \( 1 - 1.29T + 8T^{2} \)
3 \( 1 + 3.36T + 27T^{2} \)
5 \( 1 - 1.78T + 125T^{2} \)
7 \( 1 + 27.5T + 343T^{2} \)
11 \( 1 - 9.45T + 1.33e3T^{2} \)
13 \( 1 - 38.4T + 2.19e3T^{2} \)
17 \( 1 - 54.0T + 4.91e3T^{2} \)
19 \( 1 - 37.3T + 6.85e3T^{2} \)
23 \( 1 + 100.T + 1.21e4T^{2} \)
29 \( 1 - 175.T + 2.43e4T^{2} \)
31 \( 1 + 247.T + 2.97e4T^{2} \)
37 \( 1 - 234.T + 5.06e4T^{2} \)
41 \( 1 - 325.T + 6.89e4T^{2} \)
47 \( 1 - 148.T + 1.03e5T^{2} \)
53 \( 1 - 735.T + 1.48e5T^{2} \)
59 \( 1 + 275.T + 2.05e5T^{2} \)
61 \( 1 - 30.9T + 2.26e5T^{2} \)
67 \( 1 + 27.3T + 3.00e5T^{2} \)
71 \( 1 + 66.0T + 3.57e5T^{2} \)
73 \( 1 + 511.T + 3.89e5T^{2} \)
79 \( 1 + 1.19e3T + 4.93e5T^{2} \)
83 \( 1 + 144.T + 5.71e5T^{2} \)
89 \( 1 - 769.T + 7.04e5T^{2} \)
97 \( 1 - 1.88e3T + 9.12e5T^{2} \)
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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

−8.673389213073413328841288439466, −7.68531227034512394330319354682, −6.52604737972396470513358450742, −5.78607611759116594581513606574, −5.64406477233472084603088224442, −4.25021572807759312746108034957, −3.57621553424830592450677373702, −2.74192738426718430821686019131, −0.925377485845133948859926572905, 0, 0.925377485845133948859926572905, 2.74192738426718430821686019131, 3.57621553424830592450677373702, 4.25021572807759312746108034957, 5.64406477233472084603088224442, 5.78607611759116594581513606574, 6.52604737972396470513358450742, 7.68531227034512394330319354682, 8.673389213073413328841288439466

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