Properties

Label 2-3744-416.363-c0-0-2
Degree $2$
Conductor $3744$
Sign $0.923 - 0.382i$
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  + (0.980 + 0.195i)2-s + (0.923 + 0.382i)4-s + (−0.360 − 0.149i)5-s + (0.831 + 0.555i)8-s + (−0.324 − 0.216i)10-s + (1.53 + 0.636i)11-s + (−0.382 − 0.923i)13-s + (0.707 + 0.707i)16-s + (−0.275 − 0.275i)20-s + (1.38 + 0.923i)22-s + (−0.599 − 0.599i)25-s + (−0.195 − 0.980i)26-s + (0.555 + 0.831i)32-s + (−0.216 − 0.324i)40-s + (1.38 + 1.38i)41-s + ⋯
L(s)  = 1  + (0.980 + 0.195i)2-s + (0.923 + 0.382i)4-s + (−0.360 − 0.149i)5-s + (0.831 + 0.555i)8-s + (−0.324 − 0.216i)10-s + (1.53 + 0.636i)11-s + (−0.382 − 0.923i)13-s + (0.707 + 0.707i)16-s + (−0.275 − 0.275i)20-s + (1.38 + 0.923i)22-s + (−0.599 − 0.599i)25-s + (−0.195 − 0.980i)26-s + (0.555 + 0.831i)32-s + (−0.216 − 0.324i)40-s + (1.38 + 1.38i)41-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $0.923 - 0.382i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (3691, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :0),\ 0.923 - 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.499296926\)
\(L(\frac12)\) \(\approx\) \(2.499296926\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.980 - 0.195i)T \)
3 \( 1 \)
13 \( 1 + (0.382 + 0.923i)T \)
good5 \( 1 + (0.360 + 0.149i)T + (0.707 + 0.707i)T^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + (-1.53 - 0.636i)T + (0.707 + 0.707i)T^{2} \)
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 \( 1 + (-1.38 - 1.38i)T + iT^{2} \)
43 \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \)
47 \( 1 + 1.11iT - T^{2} \)
53 \( 1 + (0.707 + 0.707i)T^{2} \)
59 \( 1 + (-0.750 + 1.81i)T + (-0.707 - 0.707i)T^{2} \)
61 \( 1 + (1.30 - 0.541i)T + (0.707 - 0.707i)T^{2} \)
67 \( 1 + (0.707 - 0.707i)T^{2} \)
71 \( 1 + (-1.17 - 1.17i)T + iT^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + 1.84T + T^{2} \)
83 \( 1 + (-0.149 - 0.360i)T + (-0.707 + 0.707i)T^{2} \)
89 \( 1 + (1.17 - 1.17i)T - iT^{2} \)
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

−8.443672260431030263402936418829, −7.921152567966950147195620919724, −7.08256576558661045417717884419, −6.48792753160794770873742926014, −5.74786910592610342230620714509, −4.82269443496610572405091224589, −4.20898524697828333037541022271, −3.47935981004837675111503827070, −2.51181809370420465778347301617, −1.38280807987676023904890400443, 1.29555610683493295857639325008, 2.29410116457381477646665250603, 3.42740159623497072288253724792, 3.98810963790651458158301887685, 4.64824790277903636479876753684, 5.75261725676690541554006445401, 6.25675629205548146007653583093, 7.11703087420924351454664769313, 7.54593948629391793997348784261, 8.811734829914625882868814788399

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