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Properties

Label 2-3744-416.155-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: $\chi_{3744} (1819, \cdot )$
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

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

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