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Properties

Label 2-756-28.27-c1-0-41
Degree $2$
Conductor $756$
Sign $1$
Analytic cond. $6.03669$
Root an. cond. $2.45696$
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  + 1.41i·2-s − 2.00·4-s − 0.250i·5-s + 2.64·7-s − 2.82i·8-s + 0.354·10-s − 4.90i·11-s + 3.74i·14-s + 4.00·16-s − 3.74i·17-s − 8.64·19-s + 0.500i·20-s + 6.93·22-s − 9.14i·23-s + 4.93·25-s + ⋯
L(s)  = 1  + 0.999i·2-s − 1.00·4-s − 0.112i·5-s + 0.999·7-s − 1.00i·8-s + 0.112·10-s − 1.47i·11-s + 1.00i·14-s + 1.00·16-s − 0.907i·17-s − 1.98·19-s + 0.112i·20-s + 1.47·22-s − 1.90i·23-s + 0.987·25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $1$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35694\)
\(L(\frac12)\) \(\approx\) \(1.35694\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
3 \( 1 \)
7 \( 1 - 2.64T \)
good5 \( 1 + 0.250iT - 5T^{2} \)
11 \( 1 + 4.90iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 3.74iT - 17T^{2} \)
19 \( 1 + 8.64T + 19T^{2} \)
23 \( 1 + 9.14iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 8.29T + 31T^{2} \)
37 \( 1 - 3.93T + 37T^{2} \)
41 \( 1 - 12.4iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 13.3iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 11.4iT - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−10.38668065111654095470209424018, −9.086626989943541855666842839905, −8.386714352442739572274280656644, −8.041101329667857670516608198995, −6.64517430272311836916293408539, −6.14069549652339257499453065994, −4.87524905566930969665458768793, −4.36632200498403689343021242560, −2.78887369159828521315549200402, −0.75695413319349394326302322762, 1.54978917119159077659978053998, 2.39490201184095677325762618730, 3.98161772102204183432411001079, 4.59381120730057052136298734104, 5.61871334706560740919154290932, 6.97958408678587891428006767534, 8.030918958619118780686932294348, 8.704441137601122418393011797563, 9.700596198901630334876992355872, 10.48914663197750947990957288276

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