Properties

Degree $2$
Conductor $11830$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 5-s + 7-s − 8-s − 3·9-s − 10-s − 4·11-s − 14-s + 16-s + 2·17-s + 3·18-s + 20-s + 4·22-s + 25-s + 28-s + 6·29-s − 8·31-s − 32-s − 2·34-s + 35-s − 3·36-s + 10·37-s − 40-s − 2·41-s + 4·43-s − 4·44-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 9-s − 0.316·10-s − 1.20·11-s − 0.267·14-s + 1/4·16-s + 0.485·17-s + 0.707·18-s + 0.223·20-s + 0.852·22-s + 1/5·25-s + 0.188·28-s + 1.11·29-s − 1.43·31-s − 0.176·32-s − 0.342·34-s + 0.169·35-s − 1/2·36-s + 1.64·37-s − 0.158·40-s − 0.312·41-s + 0.609·43-s − 0.603·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11830\)    =    \(2 \cdot 5 \cdot 7 \cdot 13^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{11830} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 11830,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 - T \)
13 \( 1 \)
good3 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−16.82910624908743, −16.22509039887183, −15.64559581526341, −15.02266102143043, −14.33373552519751, −14.08326863854865, −13.15074435000924, −12.72729000676801, −12.01919833678731, −11.16042871755002, −11.06880810070532, −10.22506052626149, −9.743004797780340, −9.086647267910902, −8.359001708426036, −8.010805133905225, −7.372960118912217, −6.539176569653221, −5.820411265218525, −5.353864723355323, −4.637309903361985, −3.463521209863706, −2.721605343890835, −2.164868888385449, −1.084266288148160, 0, 1.084266288148160, 2.164868888385449, 2.721605343890835, 3.463521209863706, 4.637309903361985, 5.353864723355323, 5.820411265218525, 6.539176569653221, 7.372960118912217, 8.010805133905225, 8.359001708426036, 9.086647267910902, 9.743004797780340, 10.22506052626149, 11.06880810070532, 11.16042871755002, 12.01919833678731, 12.72729000676801, 13.15074435000924, 14.08326863854865, 14.33373552519751, 15.02266102143043, 15.64559581526341, 16.22509039887183, 16.82910624908743

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