Properties

Label 2-1421-1.1-c3-0-96
Degree $2$
Conductor $1421$
Sign $1$
Analytic cond. $83.8417$
Root an. cond. $9.15651$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.41·2-s + 0.757·3-s − 2.17·4-s + 10.6·5-s − 1.82·6-s + 24.5·8-s − 26.4·9-s − 25.7·10-s + 39.3·11-s − 1.64·12-s − 23.7·13-s + 8.07·15-s − 41.9·16-s − 4.54·17-s + 63.7·18-s + 155.·19-s − 23.1·20-s − 94.9·22-s − 41.8·23-s + 18.5·24-s − 11.4·25-s + 57.3·26-s − 40.4·27-s + 29·29-s − 19.4·30-s + 57.9·31-s − 95.2·32-s + ⋯
L(s)  = 1  − 0.853·2-s + 0.145·3-s − 0.271·4-s + 0.953·5-s − 0.124·6-s + 1.08·8-s − 0.978·9-s − 0.813·10-s + 1.07·11-s − 0.0395·12-s − 0.507·13-s + 0.138·15-s − 0.654·16-s − 0.0648·17-s + 0.835·18-s + 1.87·19-s − 0.258·20-s − 0.920·22-s − 0.379·23-s + 0.158·24-s − 0.0914·25-s + 0.432·26-s − 0.288·27-s + 0.185·29-s − 0.118·30-s + 0.335·31-s − 0.526·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1421 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1421 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1421\)    =    \(7^{2} \cdot 29\)
Sign: $1$
Analytic conductor: \(83.8417\)
Root analytic conductor: \(9.15651\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1421,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.468605835\)
\(L(\frac12)\) \(\approx\) \(1.468605835\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
29 \( 1 - 29T \)
good2 \( 1 + 2.41T + 8T^{2} \)
3 \( 1 - 0.757T + 27T^{2} \)
5 \( 1 - 10.6T + 125T^{2} \)
11 \( 1 - 39.3T + 1.33e3T^{2} \)
13 \( 1 + 23.7T + 2.19e3T^{2} \)
17 \( 1 + 4.54T + 4.91e3T^{2} \)
19 \( 1 - 155.T + 6.85e3T^{2} \)
23 \( 1 + 41.8T + 1.21e4T^{2} \)
31 \( 1 - 57.9T + 2.97e4T^{2} \)
37 \( 1 - 235.T + 5.06e4T^{2} \)
41 \( 1 - 175.T + 6.89e4T^{2} \)
43 \( 1 + 402.T + 7.95e4T^{2} \)
47 \( 1 + 227.T + 1.03e5T^{2} \)
53 \( 1 - 673.T + 1.48e5T^{2} \)
59 \( 1 - 800.T + 2.05e5T^{2} \)
61 \( 1 - 222.T + 2.26e5T^{2} \)
67 \( 1 + 524.T + 3.00e5T^{2} \)
71 \( 1 + 281.T + 3.57e5T^{2} \)
73 \( 1 + 1.22e3T + 3.89e5T^{2} \)
79 \( 1 - 611.T + 4.93e5T^{2} \)
83 \( 1 + 515.T + 5.71e5T^{2} \)
89 \( 1 - 358.T + 7.04e5T^{2} \)
97 \( 1 + 829.T + 9.12e5T^{2} \)
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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

−9.276711027024936482688989358161, −8.576319601095259001868918791479, −7.76378137545148771228603598276, −6.88336093772848305249775681240, −5.84828577779909208480906640299, −5.16973407631350342054151336092, −4.02832143714565877389367044259, −2.84264723099625218206690757567, −1.69913191975351775496309056745, −0.70966006069231470176172940773, 0.70966006069231470176172940773, 1.69913191975351775496309056745, 2.84264723099625218206690757567, 4.02832143714565877389367044259, 5.16973407631350342054151336092, 5.84828577779909208480906640299, 6.88336093772848305249775681240, 7.76378137545148771228603598276, 8.576319601095259001868918791479, 9.276711027024936482688989358161

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