Properties

Label 2-209e2-1.1-c1-0-8
Degree $2$
Conductor $43681$
Sign $-1$
Analytic cond. $348.794$
Root an. cond. $18.6760$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2·3-s − 4-s + 5-s − 2·6-s + 2·7-s − 3·8-s + 9-s + 10-s + 2·12-s + 13-s + 2·14-s − 2·15-s − 16-s + 5·17-s + 18-s − 20-s − 4·21-s + 2·23-s + 6·24-s − 4·25-s + 26-s + 4·27-s − 2·28-s + 9·29-s − 2·30-s + 2·31-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s − 1/2·4-s + 0.447·5-s − 0.816·6-s + 0.755·7-s − 1.06·8-s + 1/3·9-s + 0.316·10-s + 0.577·12-s + 0.277·13-s + 0.534·14-s − 0.516·15-s − 1/4·16-s + 1.21·17-s + 0.235·18-s − 0.223·20-s − 0.872·21-s + 0.417·23-s + 1.22·24-s − 4/5·25-s + 0.196·26-s + 0.769·27-s − 0.377·28-s + 1.67·29-s − 0.365·30-s + 0.359·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43681 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43681 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43681\)    =    \(11^{2} \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(348.794\)
Root analytic conductor: \(18.6760\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{43681} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 43681,\ (\ :1/2),\ -1)\)

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)$
bad11 \( 1 \)
19 \( 1 \)
good2 \( 1 - T + p T^{2} \)
3 \( 1 + 2 T + p T^{2} \)
5 \( 1 - T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−14.65971922265976, −14.40867344801262, −13.87047669250219, −13.49079680447981, −12.77886431272953, −12.33669489030356, −11.84424675423949, −11.53354455083072, −10.90117530826619, −10.14997767064946, −10.01515903947925, −9.117229409850715, −8.637755575173094, −7.993864806724421, −7.478008567507343, −6.405420897603602, −6.218924384071943, −5.658299454087098, −5.019139449964232, −4.788557233104521, −4.134879869239678, −3.235380587648730, −2.766916759029288, −1.563079450496731, −0.9783039804051472, 0, 0.9783039804051472, 1.563079450496731, 2.766916759029288, 3.235380587648730, 4.134879869239678, 4.788557233104521, 5.019139449964232, 5.658299454087098, 6.218924384071943, 6.405420897603602, 7.478008567507343, 7.993864806724421, 8.637755575173094, 9.117229409850715, 10.01515903947925, 10.14997767064946, 10.90117530826619, 11.53354455083072, 11.84424675423949, 12.33669489030356, 12.77886431272953, 13.49079680447981, 13.87047669250219, 14.40867344801262, 14.65971922265976

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