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Properties

Label 2-756-28.19-c1-0-10
Degree $2$
Conductor $756$
Sign $0.877 - 0.478i$
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  + (0.0660 − 1.41i)2-s + (−1.99 − 0.186i)4-s + (−1.13 − 0.652i)5-s + (−2.50 − 0.851i)7-s + (−0.394 + 2.80i)8-s + (−0.996 + 1.55i)10-s + (−1.66 + 0.960i)11-s + 4.88i·13-s + (−1.36 + 3.48i)14-s + (3.93 + 0.742i)16-s + (7.03 − 4.06i)17-s + (−2.90 + 5.03i)19-s + (2.12 + 1.51i)20-s + (1.24 + 2.41i)22-s + (7.15 + 4.13i)23-s + ⋯
L(s)  = 1  + (0.0466 − 0.998i)2-s + (−0.995 − 0.0932i)4-s + (−0.505 − 0.291i)5-s + (−0.946 − 0.321i)7-s + (−0.139 + 0.990i)8-s + (−0.315 + 0.491i)10-s + (−0.501 + 0.289i)11-s + 1.35i·13-s + (−0.365 + 0.930i)14-s + (0.982 + 0.185i)16-s + (1.70 − 0.985i)17-s + (−0.666 + 1.15i)19-s + (0.476 + 0.337i)20-s + (0.265 + 0.514i)22-s + (1.49 + 0.861i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 - 0.478i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.877 - 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.877 - 0.478i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.877 - 0.478i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.652297 + 0.166322i\)
\(L(\frac12)\) \(\approx\) \(0.652297 + 0.166322i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0660 + 1.41i)T \)
3 \( 1 \)
7 \( 1 + (2.50 + 0.851i)T \)
good5 \( 1 + (1.13 + 0.652i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.66 - 0.960i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.88iT - 13T^{2} \)
17 \( 1 + (-7.03 + 4.06i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.90 - 5.03i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-7.15 - 4.13i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.41T + 29T^{2} \)
31 \( 1 + (0.682 + 1.18i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.49 - 9.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 4.57iT - 41T^{2} \)
43 \( 1 + 0.547iT - 43T^{2} \)
47 \( 1 + (2.44 - 4.22i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.31 - 4.01i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.07 + 5.31i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-8.77 - 5.06i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.85 - 3.95i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.66iT - 71T^{2} \)
73 \( 1 + (7.64 - 4.41i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.79 + 3.92i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.23T + 83T^{2} \)
89 \( 1 + (1.97 + 1.14i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 2.86iT - 97T^{2} \)
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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

−10.24305416541276089834200997318, −9.776858693623566461839669038756, −8.984664678556286677162313009244, −7.950001303072594767901877615330, −7.06874775520802111812031515222, −5.76585536662025292494319678450, −4.72281024093737653310088051336, −3.77285881008831752831037533589, −2.92075497948639909215524249587, −1.34809881700796779952909246705, 0.37112086701108510973279616076, 3.03560747427923967639861781274, 3.71953028978508621910871080881, 5.24481414157834916097410761002, 5.78135321566149869781844293416, 6.88617810155294606367803364919, 7.57177409835057922045144945284, 8.451735334176294095057984496043, 9.179957785850607659230587711955, 10.27190513772417240097387889616

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