Properties

Label 2-116886-1.1-c1-0-38
Degree $2$
Conductor $116886$
Sign $-1$
Analytic cond. $933.339$
Root an. cond. $30.5506$
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 + 4·5-s + 6-s + 7-s − 8-s + 9-s − 4·10-s − 12-s + 6·13-s − 14-s − 4·15-s + 16-s − 2·17-s − 18-s + 6·19-s + 4·20-s − 21-s − 23-s + 24-s + 11·25-s − 6·26-s − 27-s + 28-s − 8·29-s + 4·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.78·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 1.26·10-s − 0.288·12-s + 1.66·13-s − 0.267·14-s − 1.03·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 1.37·19-s + 0.894·20-s − 0.218·21-s − 0.208·23-s + 0.204·24-s + 11/5·25-s − 1.17·26-s − 0.192·27-s + 0.188·28-s − 1.48·29-s + 0.730·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(116886\)    =    \(2 \cdot 3 \cdot 7 \cdot 11^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(933.339\)
Root analytic conductor: \(30.5506\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{116886} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 116886,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
11 \( 1 \)
23 \( 1 + T \)
good5 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 15 T + p T^{2} \)
71 \( 1 + 7 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 9 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
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

−13.72491646326901, −13.42133226568826, −12.96631988645857, −12.37366523021837, −11.66299903325266, −11.35830189018165, −10.76309068395830, −10.42154514472304, −10.03151020679938, −9.290003290565140, −9.111744051230801, −8.639216131446301, −7.944503043623480, −7.242361496648467, −6.899562914132864, −6.165552421623265, −5.788161228231078, −5.583477630612265, −4.884615148237680, −4.131517122940000, −3.357328730878954, −2.762245014217288, −1.925538393294360, −1.445720690268672, −1.160525604671109, 0, 1.160525604671109, 1.445720690268672, 1.925538393294360, 2.762245014217288, 3.357328730878954, 4.131517122940000, 4.884615148237680, 5.583477630612265, 5.788161228231078, 6.165552421623265, 6.899562914132864, 7.242361496648467, 7.944503043623480, 8.639216131446301, 9.111744051230801, 9.290003290565140, 10.03151020679938, 10.42154514472304, 10.76309068395830, 11.35830189018165, 11.66299903325266, 12.37366523021837, 12.96631988645857, 13.42133226568826, 13.72491646326901

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