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Properties

Label 2-756-63.38-c1-0-3
Degree $2$
Conductor $756$
Sign $0.999 + 0.00179i$
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  + (−1.09 + 1.89i)5-s + (−1.25 − 2.32i)7-s + (1.26 − 0.732i)11-s + (2.92 − 1.69i)13-s + (−1.32 + 2.28i)17-s + (6.87 − 3.97i)19-s + (3.47 + 2.00i)23-s + (0.117 + 0.203i)25-s + (6.71 + 3.87i)29-s + 0.706i·31-s + (5.77 + 0.160i)35-s + (1.41 + 2.45i)37-s + (−3.74 − 6.48i)41-s + (−1.27 + 2.20i)43-s + 12.5·47-s + ⋯
L(s)  = 1  + (−0.488 + 0.845i)5-s + (−0.475 − 0.879i)7-s + (0.382 − 0.220i)11-s + (0.811 − 0.468i)13-s + (−0.320 + 0.555i)17-s + (1.57 − 0.911i)19-s + (0.724 + 0.418i)23-s + (0.0234 + 0.0406i)25-s + (1.24 + 0.719i)29-s + 0.126i·31-s + (0.975 + 0.0271i)35-s + (0.233 + 0.403i)37-s + (−0.584 − 1.01i)41-s + (−0.193 + 0.335i)43-s + 1.83·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00179i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00179i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.999 + 0.00179i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.999 + 0.00179i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42977 - 0.00128662i\)
\(L(\frac12)\) \(\approx\) \(1.42977 - 0.00128662i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.25 + 2.32i)T \)
good5 \( 1 + (1.09 - 1.89i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.26 + 0.732i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.92 + 1.69i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.32 - 2.28i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.87 + 3.97i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.47 - 2.00i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-6.71 - 3.87i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.706iT - 31T^{2} \)
37 \( 1 + (-1.41 - 2.45i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.74 + 6.48i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.27 - 2.20i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 12.5T + 47T^{2} \)
53 \( 1 + (-2.41 - 1.39i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 13.4T + 59T^{2} \)
61 \( 1 + 7.79iT - 61T^{2} \)
67 \( 1 - 5.84T + 67T^{2} \)
71 \( 1 + 11.6iT - 71T^{2} \)
73 \( 1 + (3.95 + 2.28i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 9.38T + 79T^{2} \)
83 \( 1 + (1.70 - 2.95i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.61 + 8.00i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.38 - 3.68i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.63557729221620104009350504239, −9.527936272936279585735292111529, −8.657722190449758732617589185725, −7.51806007675769969798362220285, −6.97781415173774144735130916460, −6.12451613296528886483047283050, −4.84274621218702800202429813004, −3.56885428635930651487407801238, −3.08412698892047045350487933295, −1.01740971604093126688726942788, 1.09180279125250819075131966544, 2.72878044112212623026590485470, 3.93133403914788674861659156729, 4.91863050748836188021168354938, 5.87898879940786578535484518342, 6.80179065881503093526730033354, 7.918898185663389137885420898503, 8.767362239413135118987986506740, 9.290481644932708692885768950511, 10.20257952980086261201351821861

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