Properties

Label 2-22386-1.1-c1-0-25
Degree $2$
Conductor $22386$
Sign $-1$
Analytic cond. $178.753$
Root an. cond. $13.3698$
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 + 3-s + 4-s + 6-s + 7-s + 8-s + 9-s + 12-s + 13-s + 14-s + 16-s − 6·17-s + 18-s − 4·19-s + 21-s + 24-s − 5·25-s + 26-s + 27-s + 28-s + 6·29-s − 4·31-s + 32-s − 6·34-s + 36-s + 2·37-s − 4·38-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.288·12-s + 0.277·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.917·19-s + 0.218·21-s + 0.204·24-s − 25-s + 0.196·26-s + 0.192·27-s + 0.188·28-s + 1.11·29-s − 0.718·31-s + 0.176·32-s − 1.02·34-s + 1/6·36-s + 0.328·37-s − 0.648·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(22386\)    =    \(2 \cdot 3 \cdot 7 \cdot 13 \cdot 41\)
Sign: $-1$
Analytic conductor: \(178.753\)
Root analytic conductor: \(13.3698\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 22386,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 - T \)
3 \( 1 - T \)
7 \( 1 - T \)
13 \( 1 - T \)
41 \( 1 + T \)
good5 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−15.64544263932852, −15.19479800168627, −14.68316658117965, −14.11059201919170, −13.66564396805786, −13.13289330209126, −12.73151623446536, −12.03311742173739, −11.43159115808876, −10.95624344407962, −10.40191043305114, −9.746197554744651, −8.990636678403984, −8.538879946870052, −7.956073916709192, −7.328874966948356, −6.572361096151101, −6.239830889975726, −5.382907272327789, −4.603235818263475, −4.245219636934777, −3.540357619780032, −2.715647435777059, −2.097585733680272, −1.430836187473806, 0, 1.430836187473806, 2.097585733680272, 2.715647435777059, 3.540357619780032, 4.245219636934777, 4.603235818263475, 5.382907272327789, 6.239830889975726, 6.572361096151101, 7.328874966948356, 7.956073916709192, 8.538879946870052, 8.990636678403984, 9.746197554744651, 10.40191043305114, 10.95624344407962, 11.43159115808876, 12.03311742173739, 12.73151623446536, 13.13289330209126, 13.66564396805786, 14.11059201919170, 14.68316658117965, 15.19479800168627, 15.64544263932852

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