Properties

Label 2-3744-13.5-c0-0-3
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

\(L(\frac{1}{2})\) \(\approx\) \(1.600915766\)
\(L(\frac12)\) \(\approx\) \(1.600915766\)
\(L(1)\) 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

−8.763181695814740425332622879866, −8.077196883974705767795534515826, −7.06536350557419186549458761571, −6.29981781133940215906406851320, −5.70167595478176793866327755513, −4.74554718864563633975788540722, −4.34554960100854779785093159264, −2.99740109841387028626714697373, −2.00928541277879096433162305045, −1.12451573249837154139486027186, 1.25156413959512857645501907717, 2.64578695447521666299827688073, 2.87017453069992817163761397677, 4.15017008628009042565288530091, 5.08644100637190729889763225675, 6.05006659469721442765779364500, 6.28538977692793145722632479705, 7.28467137467087155827139534592, 7.914363884419773255436589195893, 8.782447344429322762409291767914

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