Properties

Label 2-756-28.3-c1-0-0
Degree $2$
Conductor $756$
Sign $-0.221 - 0.975i$
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  + (0.691 − 1.23i)2-s + (−1.04 − 1.70i)4-s + (−0.766 + 0.442i)5-s + (−2.19 + 1.48i)7-s + (−2.82 + 0.108i)8-s + (0.0160 + 1.25i)10-s + (−1.90 − 1.10i)11-s − 0.687i·13-s + (0.313 + 3.72i)14-s + (−1.81 + 3.56i)16-s + (−0.0941 − 0.0543i)17-s + (0.801 + 1.38i)19-s + (1.55 + 0.845i)20-s + (−2.68 + 1.59i)22-s + (−6.16 + 3.56i)23-s + ⋯
L(s)  = 1  + (0.488 − 0.872i)2-s + (−0.522 − 0.852i)4-s + (−0.342 + 0.197i)5-s + (−0.828 + 0.560i)7-s + (−0.999 + 0.0385i)8-s + (0.00508 + 0.395i)10-s + (−0.575 − 0.332i)11-s − 0.190i·13-s + (0.0838 + 0.996i)14-s + (−0.454 + 0.890i)16-s + (−0.0228 − 0.0131i)17-s + (0.183 + 0.318i)19-s + (0.347 + 0.189i)20-s + (−0.571 + 0.339i)22-s + (−1.28 + 0.742i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.221 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.221 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.221 - 0.975i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.221 - 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.108653 + 0.136126i\)
\(L(\frac12)\) \(\approx\) \(0.108653 + 0.136126i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.691 + 1.23i)T \)
3 \( 1 \)
7 \( 1 + (2.19 - 1.48i)T \)
good5 \( 1 + (0.766 - 0.442i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.90 + 1.10i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 0.687iT - 13T^{2} \)
17 \( 1 + (0.0941 + 0.0543i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.801 - 1.38i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.16 - 3.56i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.82T + 29T^{2} \)
31 \( 1 + (3.76 - 6.51i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.04 - 7.00i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 8.74iT - 41T^{2} \)
43 \( 1 + 2.09iT - 43T^{2} \)
47 \( 1 + (6.62 + 11.4i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.78 + 6.54i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.23 - 5.60i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.33 + 0.772i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.46 + 1.99i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.352iT - 71T^{2} \)
73 \( 1 + (-0.816 - 0.471i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.08 - 2.35i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.80T + 83T^{2} \)
89 \( 1 + (-11.7 + 6.76i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.8iT - 97T^{2} \)
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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

−10.56012939731021122562908124545, −9.946215548962082372933990162958, −9.113994985232707301585882252703, −8.186875794995063333400013628489, −6.99287682848353525858953767121, −5.83739558457387872438651708610, −5.28223672445037798581334190473, −3.79954751765836948392856682656, −3.19171125317899566597498596375, −1.93856751111280441035173311952, 0.07074412336536973318955531953, 2.64310468800940614232297989515, 3.92029394388752596066365476194, 4.52937155494569025464755644682, 5.82493458203961040486867989580, 6.49066639146654761561625833563, 7.59050275065096269421121538062, 7.978420455818847489800394845825, 9.242239675865899192729775524230, 9.846079045537950146981469636159

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