Properties

Degree $2$
Conductor $3744$
Sign $0.173 - 0.984i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.173 + 0.984i)3-s + (−0.173 + 0.300i)5-s + (−0.939 − 1.62i)7-s + (−0.939 − 0.342i)9-s + (−0.5 + 0.866i)13-s + (−0.266 − 0.223i)15-s + 1.53·17-s + (1.76 − 0.642i)21-s + (0.439 + 0.761i)25-s + (0.5 − 0.866i)27-s + (−0.5 + 0.866i)31-s + 0.652·35-s + 0.347·37-s + (−0.766 − 0.642i)39-s + (0.766 + 1.32i)43-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)3-s + (−0.173 + 0.300i)5-s + (−0.939 − 1.62i)7-s + (−0.939 − 0.342i)9-s + (−0.5 + 0.866i)13-s + (−0.266 − 0.223i)15-s + 1.53·17-s + (1.76 − 0.642i)21-s + (0.439 + 0.761i)25-s + (0.5 − 0.866i)27-s + (−0.5 + 0.866i)31-s + 0.652·35-s + 0.347·37-s + (−0.766 − 0.642i)39-s + (0.766 + 1.32i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $0.173 - 0.984i$
Motivic weight: \(0\)
Character: $\chi_{3744} (1039, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :0),\ 0.173 - 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9027350377\)
\(L(\frac12)\) \(\approx\) \(0.9027350377\)
\(L(1)\) 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 \( 1 + (0.173 - 0.984i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.939 + 1.62i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 - 1.53T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 - 0.347T + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - 1.87T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−9.231622239530103967789636195657, −7.989389754691687230081687714829, −7.33668810728480254308415327643, −6.69276897530970450303748510276, −5.89021481091345109066053786379, −4.91891875610524054993050614658, −4.16360597452696556417884068996, −3.52115451161312057333861382525, −2.89344185370798417512906918267, −1.07628345669822305692950463093, 0.63111814297088148890816658048, 2.15790323934057622335037041346, 2.77427477467094565633722405222, 3.63843085583174294992786969762, 5.21154120441001507234473902026, 5.57415475340358738537472973091, 6.19723449876003247921626565217, 7.06557001946012807835454684388, 7.84944806275273809174057428316, 8.429330540639321356516658184878

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