Properties

Label 2-3744-416.51-c0-0-1
Degree $2$
Conductor $3744$
Sign $0.382 + 0.923i$
Analytic cond. $1.86849$
Root an. cond. $1.36693$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.831 − 0.555i)2-s + (0.382 + 0.923i)4-s + (−0.425 − 1.02i)5-s + (0.195 − 0.980i)8-s + (−0.216 + 1.08i)10-s + (0.149 + 0.360i)11-s + (0.923 + 0.382i)13-s + (−0.707 + 0.707i)16-s + (0.785 − 0.785i)20-s + (0.0761 − 0.382i)22-s + (−0.165 + 0.165i)25-s + (−0.555 − 0.831i)26-s + (0.980 − 0.195i)32-s + (−1.08 + 0.216i)40-s + (1.17 − 1.17i)41-s + ⋯
L(s)  = 1  + (−0.831 − 0.555i)2-s + (0.382 + 0.923i)4-s + (−0.425 − 1.02i)5-s + (0.195 − 0.980i)8-s + (−0.216 + 1.08i)10-s + (0.149 + 0.360i)11-s + (0.923 + 0.382i)13-s + (−0.707 + 0.707i)16-s + (0.785 − 0.785i)20-s + (0.0761 − 0.382i)22-s + (−0.165 + 0.165i)25-s + (−0.555 − 0.831i)26-s + (0.980 − 0.195i)32-s + (−1.08 + 0.216i)40-s + (1.17 − 1.17i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $0.382 + 0.923i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (883, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :0),\ 0.382 + 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8304673707\)
\(L(\frac12)\) \(\approx\) \(0.8304673707\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.831 + 0.555i)T \)
3 \( 1 \)
13 \( 1 + (-0.923 - 0.382i)T \)
good5 \( 1 + (0.425 + 1.02i)T + (-0.707 + 0.707i)T^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 + (-0.149 - 0.360i)T + (-0.707 + 0.707i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + (0.707 + 0.707i)T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + (-0.707 - 0.707i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-0.707 + 0.707i)T^{2} \)
41 \( 1 + (-1.17 + 1.17i)T - iT^{2} \)
43 \( 1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2} \)
47 \( 1 - 1.96iT - T^{2} \)
53 \( 1 + (-0.707 + 0.707i)T^{2} \)
59 \( 1 + (-1.53 + 0.636i)T + (0.707 - 0.707i)T^{2} \)
61 \( 1 + (-0.541 + 1.30i)T + (-0.707 - 0.707i)T^{2} \)
67 \( 1 + (-0.707 - 0.707i)T^{2} \)
71 \( 1 + (0.275 - 0.275i)T - iT^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + 0.765T + T^{2} \)
83 \( 1 + (1.02 + 0.425i)T + (0.707 + 0.707i)T^{2} \)
89 \( 1 + (-0.275 - 0.275i)T + iT^{2} \)
97 \( 1 - T^{2} \)
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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

−8.652745005750280116693732568132, −8.046627902627532684278537832615, −7.33942802568329145584362389580, −6.50724944197546601805167795567, −5.57869655670014345614362429482, −4.38806071750495929147650956773, −4.01719755104890832071767269585, −2.88921037991098799869058885302, −1.76831084494199667875565757354, −0.838049970990197333240654532713, 1.01401520019469534485339526944, 2.35729196274910840928953140746, 3.26855434316742609680089532315, 4.19169162525324429387479727104, 5.43728534164681208549186302029, 6.02117157213833744987877379756, 6.83842614527300048888971116177, 7.27516501964253446787721405458, 8.182581350577297409295346746212, 8.606904374116594146596131563317

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