Properties

Label 2-3744-1.1-c1-0-19
Degree $2$
Conductor $3744$
Sign $1$
Analytic cond. $29.8959$
Root an. cond. $5.46772$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·7-s − 4·11-s + 13-s + 6·17-s + 6·19-s − 5·25-s + 2·29-s − 6·31-s + 10·37-s − 8·41-s + 12·43-s − 12·47-s − 3·49-s + 6·53-s + 2·61-s + 2·67-s + 8·71-s + 14·73-s − 8·77-s + 4·79-s − 8·83-s − 4·89-s + 2·91-s + 14·97-s + 18·101-s + 4·103-s + 4·107-s + ⋯
L(s)  = 1  + 0.755·7-s − 1.20·11-s + 0.277·13-s + 1.45·17-s + 1.37·19-s − 25-s + 0.371·29-s − 1.07·31-s + 1.64·37-s − 1.24·41-s + 1.82·43-s − 1.75·47-s − 3/7·49-s + 0.824·53-s + 0.256·61-s + 0.244·67-s + 0.949·71-s + 1.63·73-s − 0.911·77-s + 0.450·79-s − 0.878·83-s − 0.423·89-s + 0.209·91-s + 1.42·97-s + 1.79·101-s + 0.394·103-s + 0.386·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(29.8959\)
Root analytic conductor: \(5.46772\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3744} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.071615770\)
\(L(\frac12)\) \(\approx\) \(2.071615770\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 - T \)
good5 \( 1 + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
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

−8.219996968835028140380697915095, −7.83426916951893550059283692867, −7.33089646644279915863076635078, −6.12903054941914866220339826086, −5.38960155116211685922184828572, −4.96232371176504035102529879261, −3.79501225342647829970241386004, −3.02121239062718425833018276654, −1.96933794516781596753638570998, −0.861461516265349207678789990615, 0.861461516265349207678789990615, 1.96933794516781596753638570998, 3.02121239062718425833018276654, 3.79501225342647829970241386004, 4.96232371176504035102529879261, 5.38960155116211685922184828572, 6.12903054941914866220339826086, 7.33089646644279915863076635078, 7.83426916951893550059283692867, 8.219996968835028140380697915095

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