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Properties

Label 2-756-63.20-c1-0-2
Degree $2$
Conductor $756$
Sign $0.970 - 0.239i$
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  + (−0.266 − 0.462i)5-s + (2.54 + 0.715i)7-s + (3.39 + 1.96i)11-s + (−0.116 + 0.0674i)13-s − 4.32·17-s + 2.22i·19-s + (1.70 − 0.983i)23-s + (2.35 − 4.08i)25-s + (5.16 + 2.98i)29-s + (−0.800 + 0.462i)31-s + (−0.348 − 1.36i)35-s + 7.79·37-s + (4.59 + 7.95i)41-s + (3.24 − 5.62i)43-s + (3.04 − 5.27i)47-s + ⋯
L(s)  = 1  + (−0.119 − 0.206i)5-s + (0.962 + 0.270i)7-s + (1.02 + 0.591i)11-s + (−0.0324 + 0.0187i)13-s − 1.04·17-s + 0.511i·19-s + (0.355 − 0.205i)23-s + (0.471 − 0.816i)25-s + (0.959 + 0.553i)29-s + (−0.143 + 0.0829i)31-s + (−0.0589 − 0.231i)35-s + 1.28·37-s + (0.716 + 1.24i)41-s + (0.494 − 0.857i)43-s + (0.443 − 0.768i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.239i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.970 - 0.239i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (629, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.970 - 0.239i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.72113 + 0.209491i\)
\(L(\frac12)\) \(\approx\) \(1.72113 + 0.209491i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (-2.54 - 0.715i)T \)
good5 \( 1 + (0.266 + 0.462i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.39 - 1.96i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.116 - 0.0674i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.32T + 17T^{2} \)
19 \( 1 - 2.22iT - 19T^{2} \)
23 \( 1 + (-1.70 + 0.983i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.16 - 2.98i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.800 - 0.462i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.79T + 37T^{2} \)
41 \( 1 + (-4.59 - 7.95i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.24 + 5.62i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.04 + 5.27i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 11.0iT - 53T^{2} \)
59 \( 1 + (-1.89 - 3.28i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (9.35 + 5.39i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.75 + 9.97i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.22iT - 71T^{2} \)
73 \( 1 - 0.381iT - 73T^{2} \)
79 \( 1 + (4.60 - 7.97i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.28 + 2.21i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 17.1T + 89T^{2} \)
97 \( 1 + (-13.6 - 7.89i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.49729834999605687008778683779, −9.339787474326435831879752363508, −8.732002242927834733874363415809, −7.87526543317123696517100862244, −6.88772858904505457819280138883, −6.00542098482472176133556439864, −4.72013500463363053970832441656, −4.22482556978361682317838632280, −2.58798613101505544452660265977, −1.33078863482834445914291166834, 1.12630523540644738076186926846, 2.60330270864079109081789468978, 3.96595685482013970079858148038, 4.74325041573150638572982306053, 5.94222559882935992965951637930, 6.87134834207862247354377953600, 7.69663815162287893941287199609, 8.705833632177596756175027048278, 9.266743556190333999521106007566, 10.48938687170574286581642181340

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