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  + 3.88·2-s − 9.09·3-s + 7.12·4-s + 19.0·5-s − 35.3·6-s − 18.7·7-s − 3.41·8-s + 55.7·9-s + 74.0·10-s + 4.37·11-s − 64.7·12-s + 45.6·13-s − 73.0·14-s − 173.·15-s − 70.2·16-s − 62.8·17-s + 216.·18-s − 33.1·19-s + 135.·20-s + 170.·21-s + 17.0·22-s − 32.0·23-s + 31.0·24-s + 237.·25-s + 177.·26-s − 261.·27-s − 133.·28-s + ⋯
L(s)  = 1  + 1.37·2-s − 1.75·3-s + 0.890·4-s + 1.70·5-s − 2.40·6-s − 1.01·7-s − 0.150·8-s + 2.06·9-s + 2.34·10-s + 0.119·11-s − 1.55·12-s + 0.973·13-s − 1.39·14-s − 2.98·15-s − 1.09·16-s − 0.896·17-s + 2.83·18-s − 0.399·19-s + 1.51·20-s + 1.77·21-s + 0.164·22-s − 0.290·23-s + 0.264·24-s + 1.90·25-s + 1.33·26-s − 1.86·27-s − 0.903·28-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 \( 1 - 3.88T + 8T^{2} \)
3 \( 1 + 9.09T + 27T^{2} \)
5 \( 1 - 19.0T + 125T^{2} \)
7 \( 1 + 18.7T + 343T^{2} \)
11 \( 1 - 4.37T + 1.33e3T^{2} \)
13 \( 1 - 45.6T + 2.19e3T^{2} \)
17 \( 1 + 62.8T + 4.91e3T^{2} \)
19 \( 1 + 33.1T + 6.85e3T^{2} \)
23 \( 1 + 32.0T + 1.21e4T^{2} \)
29 \( 1 - 291.T + 2.43e4T^{2} \)
31 \( 1 + 83.9T + 2.97e4T^{2} \)
37 \( 1 - 90.1T + 5.06e4T^{2} \)
41 \( 1 + 422.T + 6.89e4T^{2} \)
47 \( 1 + 363.T + 1.03e5T^{2} \)
53 \( 1 - 387.T + 1.48e5T^{2} \)
59 \( 1 + 248.T + 2.05e5T^{2} \)
61 \( 1 + 64.5T + 2.26e5T^{2} \)
67 \( 1 - 80.6T + 3.00e5T^{2} \)
71 \( 1 + 936.T + 3.57e5T^{2} \)
73 \( 1 - 651.T + 3.89e5T^{2} \)
79 \( 1 - 1.11e3T + 4.93e5T^{2} \)
83 \( 1 - 26.3T + 5.71e5T^{2} \)
89 \( 1 - 347.T + 7.04e5T^{2} \)
97 \( 1 - 291.T + 9.12e5T^{2} \)
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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

−8.710719622702030547745954071802, −6.75871218259972676347426464117, −6.37503820488534195700917580506, −6.17039147021216411473488276466, −5.28416720304941409249240613151, −4.74775067280103420378379258186, −3.70499821880106567522438500038, −2.52456235382069001772238160403, −1.35857918767210711064365276868, 0, 1.35857918767210711064365276868, 2.52456235382069001772238160403, 3.70499821880106567522438500038, 4.74775067280103420378379258186, 5.28416720304941409249240613151, 6.17039147021216411473488276466, 6.37503820488534195700917580506, 6.75871218259972676347426464117, 8.710719622702030547745954071802

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