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Properties

Label 2-3744-416.51-c0-0-2
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

\(L(\frac{1}{2})\) \(\approx\) \(2.270522149\)
\(L(\frac12)\) \(\approx\) \(2.270522149\)
\(L(1)\) 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

−8.680005878891332003569572090368, −8.080144878780253889766356917140, −7.18681413869966333590534913196, −6.56865946225122800598505709484, −6.07704808733126480331261628387, −5.30772126741900960153792354949, −4.35049088652900299597035853804, −3.47015864185471645079778735052, −2.85289025638560858556788568979, −1.82185212663128567394793473161, 1.04510462627879012596324314407, 1.88925139211463666310104043789, 2.97653696816861252101273100725, 3.89851207179381682992725107097, 4.67293571538905573562391585550, 5.36175202258576186053064350060, 5.94828265521725251524210868965, 6.75918185405742824134624290243, 7.71912768754222438142749606638, 8.652507498861206010956391466655

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