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Properties

Label 2-756-28.3-c1-0-15
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

\(L(1)\) \(\approx\) \(0.649013 + 0.813115i\)
\(L(\frac12)\) \(\approx\) \(0.649013 + 0.813115i\)
\(L(\frac{3}{2})\) 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

−10.26052692252141745367924005144, −9.464061838599112809314762354202, −9.006587907836634071640026850323, −8.059979794112069452159277724588, −6.97354080858233578196304639516, −6.33568550808828923926075575995, −5.46094508308919925906143191234, −4.50274193453432275574792616972, −3.00016671903147898925968678053, −1.31190520074115376466683393894, 0.71516497867640235824376753026, 2.28159327254496380898208237769, 3.42199788474560478787745174778, 4.21841480196920841386841575572, 5.63392016452152573370200489445, 6.82278471455468814964134856936, 7.50729333354378392740665268301, 8.752726083482446086922272994204, 9.377105490395810432181363527338, 10.08408279353271356858441151189

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