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Properties

Label 2-756-28.27-c1-0-49
Degree $2$
Conductor $756$
Sign $0.624 + 0.781i$
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  + (0.748 + 1.19i)2-s + (−0.879 + 1.79i)4-s − 3.90i·5-s + (−1.12 + 2.39i)7-s + (−2.81 + 0.288i)8-s + (4.69 − 2.92i)10-s − 5.01i·11-s − 5.59i·13-s + (−3.71 + 0.435i)14-s + (−2.45 − 3.15i)16-s + 2.41i·17-s − 3.11·19-s + (7.02 + 3.43i)20-s + (6.02 − 3.75i)22-s − 3.45i·23-s + ⋯
L(s)  = 1  + (0.529 + 0.848i)2-s + (−0.439 + 0.898i)4-s − 1.74i·5-s + (−0.426 + 0.904i)7-s + (−0.994 + 0.102i)8-s + (1.48 − 0.925i)10-s − 1.51i·11-s − 1.55i·13-s + (−0.993 + 0.116i)14-s + (−0.613 − 0.789i)16-s + 0.584i·17-s − 0.714·19-s + (1.57 + 0.769i)20-s + (1.28 − 0.800i)22-s − 0.719i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 + 0.781i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.624 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.624 + 0.781i$
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: \((2,\ 756,\ (\ :1/2),\ 0.624 + 0.781i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21731 - 0.585306i\)
\(L(\frac12)\) \(\approx\) \(1.21731 - 0.585306i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.748 - 1.19i)T \)
3 \( 1 \)
7 \( 1 + (1.12 - 2.39i)T \)
good5 \( 1 + 3.90iT - 5T^{2} \)
11 \( 1 + 5.01iT - 11T^{2} \)
13 \( 1 + 5.59iT - 13T^{2} \)
17 \( 1 - 2.41iT - 17T^{2} \)
19 \( 1 + 3.11T + 19T^{2} \)
23 \( 1 + 3.45iT - 23T^{2} \)
29 \( 1 - 4.76T + 29T^{2} \)
31 \( 1 + 0.482T + 31T^{2} \)
37 \( 1 - 5.90T + 37T^{2} \)
41 \( 1 + 1.51iT - 41T^{2} \)
43 \( 1 + 1.77iT - 43T^{2} \)
47 \( 1 - 2.38T + 47T^{2} \)
53 \( 1 + 5.62T + 53T^{2} \)
59 \( 1 - 6.38T + 59T^{2} \)
61 \( 1 + 6.94iT - 61T^{2} \)
67 \( 1 - 3.92iT - 67T^{2} \)
71 \( 1 - 8.25iT - 71T^{2} \)
73 \( 1 - 3.59iT - 73T^{2} \)
79 \( 1 - 9.32iT - 79T^{2} \)
83 \( 1 + 5.76T + 83T^{2} \)
89 \( 1 + 17.6iT - 89T^{2} \)
97 \( 1 - 15.5iT - 97T^{2} \)
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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

−9.979827197497968460479368102726, −8.827587438194936281217715302392, −8.505912752487716117552560411307, −7.971173264530659233465390565394, −6.30749455173086336595983983962, −5.71367290951337978582154358050, −5.08463472656209263371593080294, −3.96493081375980725106096944995, −2.80233376572205764934009924004, −0.56865382034961860453376468225, 1.88718331343538657718957140375, 2.85720702978803380553558133470, 3.95136079026767564607754544897, 4.62442134717293213620981785508, 6.28942391066575799600072431128, 6.81216336349795009487377807120, 7.52596878511691183058784683181, 9.296468489954962606506902392230, 9.897999142891413129841935641161, 10.49442462157824556958284651171

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