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Properties

Label 2-756-28.3-c1-0-29
Degree $2$
Conductor $756$
Sign $0.938 + 0.344i$
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  + (−1.41 − 0.0181i)2-s + (1.99 + 0.0513i)4-s + (−0.766 + 0.442i)5-s + (2.19 − 1.48i)7-s + (−2.82 − 0.108i)8-s + (1.09 − 0.611i)10-s + (1.90 + 1.10i)11-s − 0.687i·13-s + (−3.12 + 2.05i)14-s + (3.99 + 0.205i)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  + (−0.999 − 0.0128i)2-s + (0.999 + 0.0256i)4-s + (−0.342 + 0.197i)5-s + (0.828 − 0.560i)7-s + (−0.999 − 0.0385i)8-s + (0.345 − 0.193i)10-s + (0.575 + 0.332i)11-s − 0.190i·13-s + (−0.835 + 0.549i)14-s + (0.998 + 0.0513i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 + 0.344i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.938 + 0.344i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.938 + 0.344i$
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: \((2,\ 756,\ (\ :1/2),\ 0.938 + 0.344i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02723 - 0.182391i\)
\(L(\frac12)\) \(\approx\) \(1.02723 - 0.182391i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.0181i)T \)
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

−10.27922883389562750709720837992, −9.406304196618644633414675175171, −8.593969866739158529629019021871, −7.69960412696444288282283456341, −7.14088079236444719482786737804, −6.17149084940892793489234448976, −4.84613619580211491962101391300, −3.67717879209532870127769156234, −2.30216011373147636600512437063, −0.936536279059162012449841916728, 1.15127299756962745842075135121, 2.41378614774702101847073905731, 3.78748530428164338347813680917, 5.15317401213766475206306800897, 6.12638027036882738578339824719, 7.15463975237318284241027418828, 7.982018771507108061147894962453, 8.757124952935238724965461492709, 9.277428552857804351620562618223, 10.39944960836784711703212256751

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