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Properties

Label 2-756-84.11-c1-0-18
Degree $2$
Conductor $756$
Sign $0.627 - 0.779i$
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  + (−1.39 − 0.230i)2-s + (1.89 + 0.642i)4-s + (−0.936 + 0.540i)5-s + (−0.749 + 2.53i)7-s + (−2.49 − 1.33i)8-s + (1.43 − 0.538i)10-s + (2.43 − 4.22i)11-s + 0.815·13-s + (1.62 − 3.36i)14-s + (3.17 + 2.43i)16-s + (−1.47 − 0.848i)17-s + (3.58 − 2.07i)19-s + (−2.12 + 0.421i)20-s + (−4.37 + 5.33i)22-s + (1.75 + 3.04i)23-s + ⋯
L(s)  = 1  + (−0.986 − 0.162i)2-s + (0.946 + 0.321i)4-s + (−0.418 + 0.241i)5-s + (−0.283 + 0.959i)7-s + (−0.881 − 0.471i)8-s + (0.452 − 0.170i)10-s + (0.735 − 1.27i)11-s + 0.226·13-s + (0.435 − 0.900i)14-s + (0.793 + 0.608i)16-s + (−0.356 − 0.205i)17-s + (0.823 − 0.475i)19-s + (−0.474 + 0.0943i)20-s + (−0.933 + 1.13i)22-s + (0.366 + 0.634i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.627 - 0.779i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.627 - 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.627 - 0.779i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.627 - 0.779i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.766457 + 0.366978i\)
\(L(\frac12)\) \(\approx\) \(0.766457 + 0.366978i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 + 0.230i)T \)
3 \( 1 \)
7 \( 1 + (0.749 - 2.53i)T \)
good5 \( 1 + (0.936 - 0.540i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.43 + 4.22i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 0.815T + 13T^{2} \)
17 \( 1 + (1.47 + 0.848i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.58 + 2.07i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.75 - 3.04i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 9.61iT - 29T^{2} \)
31 \( 1 + (-7.73 - 4.46i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.37 + 5.85i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.96iT - 41T^{2} \)
43 \( 1 - 0.510iT - 43T^{2} \)
47 \( 1 + (-3.40 - 5.90i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.99 + 1.73i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.50 - 4.34i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.85 - 11.8i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.66 + 2.11i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.8T + 71T^{2} \)
73 \( 1 + (1.49 - 2.59i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.41 + 1.97i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 11.3T + 83T^{2} \)
89 \( 1 + (-0.313 + 0.180i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 8.12T + 97T^{2} \)
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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

−10.50530461012382957926686848496, −9.244450800798235679179524783931, −9.002316188924481422312216387192, −8.089348295273037368561654223066, −7.07116785912497462140809323250, −6.30002764980171714804001016878, −5.32253487161506851074212619933, −3.51043029956096641552772091191, −2.86870410814693882520609429981, −1.20971021135349493894500798763, 0.70111705397674910282730125527, 2.13205152452124891120737422033, 3.72715097870206293175286182889, 4.67981212762962910093426713347, 6.22236075285506258962886080715, 6.87408944660656042349564116224, 7.74595057039420496504900387194, 8.388488006354963890890449635184, 9.622240862168968646440371072561, 9.927438705586936643903757907810

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