Properties

Label 2-4114-1.1-c1-0-82
Degree $2$
Conductor $4114$
Sign $-1$
Analytic cond. $32.8504$
Root an. cond. $5.73153$
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 + 2·6-s + 4·7-s − 8-s + 9-s − 2·12-s − 2·13-s − 4·14-s + 16-s + 17-s − 18-s + 4·19-s − 8·21-s + 2·24-s − 5·25-s + 2·26-s + 4·27-s + 4·28-s − 4·31-s − 32-s − 34-s + 36-s − 4·37-s − 4·38-s + 4·39-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.577·12-s − 0.554·13-s − 1.06·14-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.917·19-s − 1.74·21-s + 0.408·24-s − 25-s + 0.392·26-s + 0.769·27-s + 0.755·28-s − 0.718·31-s − 0.176·32-s − 0.171·34-s + 1/6·36-s − 0.657·37-s − 0.648·38-s + 0.640·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4114\)    =    \(2 \cdot 11^{2} \cdot 17\)
Sign: $-1$
Analytic conductor: \(32.8504\)
Root analytic conductor: \(5.73153\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{4114} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4114,\ (\ :1/2),\ -1)\)

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

−8.076455834435977550787516164955, −7.36497483724733271440784650514, −6.72084452362773307312329845344, −5.67320794510931157328652310045, −5.26894992853838733045784795081, −4.56197836266380508635038039003, −3.34611668167428925701071579672, −2.05022911292484155356003327044, −1.25284776655083263545298048613, 0, 1.25284776655083263545298048613, 2.05022911292484155356003327044, 3.34611668167428925701071579672, 4.56197836266380508635038039003, 5.26894992853838733045784795081, 5.67320794510931157328652310045, 6.72084452362773307312329845344, 7.36497483724733271440784650514, 8.076455834435977550787516164955

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