Properties

Label 2-3744-13.5-c0-0-0
Degree $2$
Conductor $3744$
Sign $-0.881 - 0.471i$
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  + (−1 + i)5-s + i·13-s i·25-s − 2·29-s + (1 + i)37-s + (−1 + i)41-s i·49-s − 2·61-s + (−1 − i)65-s + (−1 − i)73-s + (1 + i)89-s + (−1 + i)97-s + 2i·101-s + (−1 + i)109-s − 2·113-s + ⋯
L(s)  = 1  + (−1 + i)5-s + i·13-s i·25-s − 2·29-s + (1 + i)37-s + (−1 + i)41-s i·49-s − 2·61-s + (−1 − i)65-s + (−1 − i)73-s + (1 + i)89-s + (−1 + i)97-s + 2i·101-s + (−1 + i)109-s − 2·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $-0.881 - 0.471i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :0),\ -0.881 - 0.471i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5643688942\)
\(L(\frac12)\) \(\approx\) \(0.5643688942\)
\(L(1)\) 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 \( 1 \)
13 \( 1 - iT \)
good5 \( 1 + (1 - i)T - iT^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + iT^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + 2T + T^{2} \)
31 \( 1 - iT^{2} \)
37 \( 1 + (-1 - i)T + iT^{2} \)
41 \( 1 + (1 - i)T - iT^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + 2T + T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 + (1 + i)T + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (-1 - i)T + iT^{2} \)
97 \( 1 + (1 - i)T - iT^{2} \)
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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

−9.057932710982769147943482028470, −8.016814986215537215836916690020, −7.60081050113357637131959733652, −6.78302051541003112253828766859, −6.30573968906806554271417082287, −5.18708685007563075831132863235, −4.25487499357865804268424525686, −3.61202921888656863601696788705, −2.80431432439921721604670244259, −1.67573917283357882291012161368, 0.32072359653524332973986674631, 1.61508144026907618671371936935, 2.95321525711208065581558515371, 3.85431070512331280567216219998, 4.47937228504691894459332898697, 5.39546428376376608471313126149, 5.93036227048239425921511669264, 7.22151926179195700398412027215, 7.67500869256278450943271833205, 8.339742374672961279165232031769

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