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Properties

Label 2-3744-468.211-c0-0-0
Degree $2$
Conductor $3744$
Sign $0.335 - 0.941i$
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  + i·3-s + (0.5 − 0.866i)5-s − 9-s + (0.866 + 0.5i)11-s − 13-s + (0.866 + 0.5i)15-s + (−0.5 + 0.866i)17-s + (0.866 + 0.5i)19-s − i·27-s + (−0.5 + 0.866i)29-s + (0.866 + 0.5i)31-s + (−0.5 + 0.866i)33-s + (−0.5 − 0.866i)37-s − i·39-s + 2i·43-s + ⋯
L(s)  = 1  + i·3-s + (0.5 − 0.866i)5-s − 9-s + (0.866 + 0.5i)11-s − 13-s + (0.866 + 0.5i)15-s + (−0.5 + 0.866i)17-s + (0.866 + 0.5i)19-s − i·27-s + (−0.5 + 0.866i)29-s + (0.866 + 0.5i)31-s + (−0.5 + 0.866i)33-s + (−0.5 − 0.866i)37-s − i·39-s + 2i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.335 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.335 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $0.335 - 0.941i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (3487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :0),\ 0.335 - 0.941i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.329736961\)
\(L(\frac12)\) \(\approx\) \(1.329736961\)
\(L(1)\) 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 \( 1 - iT \)
13 \( 1 + T \)
good5 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - 2iT - T^{2} \)
47 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
73 \( 1 - 2T + T^{2} \)
79 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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

−8.932297551261496839671841380983, −8.398028746560395895970271306692, −7.39139348225668184477240831017, −6.52077264856988169233825216436, −5.57868874002735126391220702102, −5.07581822210537217295131316772, −4.31407651835676812276615919695, −3.60413024076639222202330569298, −2.44830285767312905372120324771, −1.34104736289256335307440789828, 0.822165747647301762884723610029, 2.25922595765585925856593337523, 2.67181343176164006913767990759, 3.72030174036234651293761655437, 4.94872543596160990770336772806, 5.75826588649149139069269700946, 6.50142330358980517652017108081, 7.05169155698371726019202753558, 7.54008083198448171073030626097, 8.534430498888288258394890354028

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