Properties

Label 2-43e2-1.1-c1-0-122
Degree $2$
Conductor $1849$
Sign $-1$
Analytic cond. $14.7643$
Root an. cond. $3.84243$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.70·2-s + 1.15·3-s + 0.891·4-s − 1.76·5-s + 1.96·6-s + 0.594·7-s − 1.88·8-s − 1.66·9-s − 2.99·10-s − 4.34·11-s + 1.03·12-s + 3.24·13-s + 1.01·14-s − 2.03·15-s − 4.98·16-s − 5.54·17-s − 2.82·18-s + 2.71·19-s − 1.57·20-s + 0.687·21-s − 7.38·22-s + 5.29·23-s − 2.17·24-s − 1.89·25-s + 5.51·26-s − 5.39·27-s + 0.529·28-s + ⋯
L(s)  = 1  + 1.20·2-s + 0.667·3-s + 0.445·4-s − 0.788·5-s + 0.802·6-s + 0.224·7-s − 0.666·8-s − 0.554·9-s − 0.948·10-s − 1.31·11-s + 0.297·12-s + 0.899·13-s + 0.270·14-s − 0.526·15-s − 1.24·16-s − 1.34·17-s − 0.666·18-s + 0.623·19-s − 0.351·20-s + 0.149·21-s − 1.57·22-s + 1.10·23-s − 0.444·24-s − 0.378·25-s + 1.08·26-s − 1.03·27-s + 0.100·28-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1849\)    =    \(43^{2}\)
Sign: $-1$
Analytic conductor: \(14.7643\)
Root analytic conductor: \(3.84243\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1849,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.70T + 2T^{2} \)
3 \( 1 - 1.15T + 3T^{2} \)
5 \( 1 + 1.76T + 5T^{2} \)
7 \( 1 - 0.594T + 7T^{2} \)
11 \( 1 + 4.34T + 11T^{2} \)
13 \( 1 - 3.24T + 13T^{2} \)
17 \( 1 + 5.54T + 17T^{2} \)
19 \( 1 - 2.71T + 19T^{2} \)
23 \( 1 - 5.29T + 23T^{2} \)
29 \( 1 + 9.86T + 29T^{2} \)
31 \( 1 - 3.73T + 31T^{2} \)
37 \( 1 - 1.34T + 37T^{2} \)
41 \( 1 + 11.2T + 41T^{2} \)
47 \( 1 - 4.08T + 47T^{2} \)
53 \( 1 + 2.41T + 53T^{2} \)
59 \( 1 + 0.735T + 59T^{2} \)
61 \( 1 - 5.11T + 61T^{2} \)
67 \( 1 + 3.84T + 67T^{2} \)
71 \( 1 + 3.83T + 71T^{2} \)
73 \( 1 + 7.45T + 73T^{2} \)
79 \( 1 - 13.3T + 79T^{2} \)
83 \( 1 - 7.79T + 83T^{2} \)
89 \( 1 - 4.50T + 89T^{2} \)
97 \( 1 - 0.184T + 97T^{2} \)
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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

−8.732068126226269779719561969399, −8.087228274343778289810630582797, −7.31021850616450077912913802994, −6.25148786750009673961654354417, −5.37876339562196670120616562666, −4.71548919792536031231560676731, −3.70370803164012896487301914929, −3.16727164204450878715260183660, −2.19825868959610866765315438797, 0, 2.19825868959610866765315438797, 3.16727164204450878715260183660, 3.70370803164012896487301914929, 4.71548919792536031231560676731, 5.37876339562196670120616562666, 6.25148786750009673961654354417, 7.31021850616450077912913802994, 8.087228274343778289810630582797, 8.732068126226269779719561969399

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