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Properties

Label 2-3744-24.11-c1-0-26
Degree $2$
Conductor $3744$
Sign $0.988 - 0.149i$
Analytic cond. $29.8959$
Root an. cond. $5.46772$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.22·5-s − 0.534i·7-s − 2.35i·11-s + i·13-s + 4.78i·17-s + 4.25·19-s − 2.55·23-s − 0.0600·25-s + 8.32·29-s + 0.284i·31-s − 1.18i·35-s + 6.30i·37-s + 4.33i·41-s + 0.666·43-s − 1.10·47-s + ⋯
L(s)  = 1  + 0.993·5-s − 0.202i·7-s − 0.710i·11-s + 0.277i·13-s + 1.16i·17-s + 0.975·19-s − 0.532·23-s − 0.0120·25-s + 1.54·29-s + 0.0510i·31-s − 0.200i·35-s + 1.03i·37-s + 0.676i·41-s + 0.101·43-s − 0.160·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.149i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.149i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $0.988 - 0.149i$
Analytic conductor: \(29.8959\)
Root analytic conductor: \(5.46772\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (2159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :1/2),\ 0.988 - 0.149i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.399213313\)
\(L(\frac12)\) \(\approx\) \(2.399213313\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.22T + 5T^{2} \)
7 \( 1 + 0.534iT - 7T^{2} \)
11 \( 1 + 2.35iT - 11T^{2} \)
17 \( 1 - 4.78iT - 17T^{2} \)
19 \( 1 - 4.25T + 19T^{2} \)
23 \( 1 + 2.55T + 23T^{2} \)
29 \( 1 - 8.32T + 29T^{2} \)
31 \( 1 - 0.284iT - 31T^{2} \)
37 \( 1 - 6.30iT - 37T^{2} \)
41 \( 1 - 4.33iT - 41T^{2} \)
43 \( 1 - 0.666T + 43T^{2} \)
47 \( 1 + 1.10T + 47T^{2} \)
53 \( 1 - 1.42T + 53T^{2} \)
59 \( 1 + 3.21iT - 59T^{2} \)
61 \( 1 + 4.98iT - 61T^{2} \)
67 \( 1 + 0.658T + 67T^{2} \)
71 \( 1 - 12.3T + 71T^{2} \)
73 \( 1 - 14.1T + 73T^{2} \)
79 \( 1 - 11.8iT - 79T^{2} \)
83 \( 1 + 14.7iT - 83T^{2} \)
89 \( 1 - 1.91iT - 89T^{2} \)
97 \( 1 + 7.69T + 97T^{2} \)
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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

−8.402972249843610364563025747425, −8.020721261261389126893899760098, −6.86054494315393861851212402652, −6.26684221187054453924828705954, −5.65773840990724816778884792808, −4.83570468373063306079499000547, −3.85091432485324019666380910403, −2.97797001084000658432401506130, −1.96165643182892232420582871004, −1.00244352395638079523561027904, 0.868722437540231590164305010549, 2.10901862874774485845925479496, 2.73040411605090913677467497965, 3.86457547931138321932072448174, 4.91970519084480657701205602064, 5.44789462651773632686385679136, 6.21782147498327825516723417966, 7.04943920887150733354684899548, 7.65389460666832138206325646617, 8.590037250546097242306057686804

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