Properties

Label 2-8112-1.1-c1-0-151
Degree $2$
Conductor $8112$
Sign $-1$
Analytic cond. $64.7746$
Root an. cond. $8.04826$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 1.04·5-s + 0.554·7-s + 9-s + 2.91·11-s + 1.04·15-s − 4.85·17-s − 0.753·19-s + 0.554·21-s − 5.76·23-s − 3.89·25-s + 27-s − 1.91·29-s − 9.51·31-s + 2.91·33-s + 0.582·35-s − 5.75·37-s − 4.91·41-s + 11.0·43-s + 1.04·45-s + 0.753·47-s − 6.69·49-s − 4.85·51-s − 7.58·53-s + 3.05·55-s − 0.753·57-s + 4.09·59-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.469·5-s + 0.209·7-s + 0.333·9-s + 0.877·11-s + 0.270·15-s − 1.17·17-s − 0.172·19-s + 0.121·21-s − 1.20·23-s − 0.779·25-s + 0.192·27-s − 0.355·29-s − 1.70·31-s + 0.506·33-s + 0.0983·35-s − 0.945·37-s − 0.767·41-s + 1.69·43-s + 0.156·45-s + 0.109·47-s − 0.956·49-s − 0.679·51-s − 1.04·53-s + 0.411·55-s − 0.0997·57-s + 0.533·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8112\)    =    \(2^{4} \cdot 3 \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(64.7746\)
Root analytic conductor: \(8.04826\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{8112} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8112,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 - T \)
13 \( 1 \)
good5 \( 1 - 1.04T + 5T^{2} \)
7 \( 1 - 0.554T + 7T^{2} \)
11 \( 1 - 2.91T + 11T^{2} \)
17 \( 1 + 4.85T + 17T^{2} \)
19 \( 1 + 0.753T + 19T^{2} \)
23 \( 1 + 5.76T + 23T^{2} \)
29 \( 1 + 1.91T + 29T^{2} \)
31 \( 1 + 9.51T + 31T^{2} \)
37 \( 1 + 5.75T + 37T^{2} \)
41 \( 1 + 4.91T + 41T^{2} \)
43 \( 1 - 11.0T + 43T^{2} \)
47 \( 1 - 0.753T + 47T^{2} \)
53 \( 1 + 7.58T + 53T^{2} \)
59 \( 1 - 4.09T + 59T^{2} \)
61 \( 1 + 3.42T + 61T^{2} \)
67 \( 1 + 1.87T + 67T^{2} \)
71 \( 1 + 10.5T + 71T^{2} \)
73 \( 1 + 10.4T + 73T^{2} \)
79 \( 1 + 1.33T + 79T^{2} \)
83 \( 1 - 2.64T + 83T^{2} \)
89 \( 1 + 9.92T + 89T^{2} \)
97 \( 1 + 17.0T + 97T^{2} \)
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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

−7.46229967118417721759462569919, −6.85468850858029517633764372090, −6.09327287000248848040981267008, −5.51142072048776315736012774662, −4.42183188683509781022088495659, −3.98182263713176879400324312065, −3.10203284691290004949931396622, −1.98238708784353215785109961113, −1.67710493702141212673445693970, 0, 1.67710493702141212673445693970, 1.98238708784353215785109961113, 3.10203284691290004949931396622, 3.98182263713176879400324312065, 4.42183188683509781022088495659, 5.51142072048776315736012774662, 6.09327287000248848040981267008, 6.85468850858029517633764372090, 7.46229967118417721759462569919

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