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Properties

Label 2-756-63.38-c1-0-7
Degree $2$
Conductor $756$
Sign $-0.111 + 0.993i$
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  + (1.48 − 2.57i)5-s + (−0.200 − 2.63i)7-s + (4.09 − 2.36i)11-s + (−3.54 + 2.04i)13-s + (0.835 − 1.44i)17-s + (−4.25 + 2.45i)19-s + (−4.25 − 2.45i)23-s + (−1.91 − 3.30i)25-s + (−0.238 − 0.137i)29-s − 1.60i·31-s + (−7.08 − 3.40i)35-s + (−1.69 − 2.93i)37-s + (3.55 + 6.15i)41-s + (5.22 − 9.05i)43-s + 10.9·47-s + ⋯
L(s)  = 1  + (0.664 − 1.15i)5-s + (−0.0756 − 0.997i)7-s + (1.23 − 0.712i)11-s + (−0.981 + 0.566i)13-s + (0.202 − 0.350i)17-s + (−0.975 + 0.563i)19-s + (−0.886 − 0.511i)23-s + (−0.382 − 0.661i)25-s + (−0.0442 − 0.0255i)29-s − 0.287i·31-s + (−1.19 − 0.575i)35-s + (−0.278 − 0.483i)37-s + (0.555 + 0.961i)41-s + (0.797 − 1.38i)43-s + 1.60·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.111 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.111 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.111 + 0.993i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.111 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04700 - 1.17067i\)
\(L(\frac12)\) \(\approx\) \(1.04700 - 1.17067i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (0.200 + 2.63i)T \)
good5 \( 1 + (-1.48 + 2.57i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.09 + 2.36i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.54 - 2.04i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.835 + 1.44i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.25 - 2.45i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.25 + 2.45i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.238 + 0.137i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.60iT - 31T^{2} \)
37 \( 1 + (1.69 + 2.93i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.55 - 6.15i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.22 + 9.05i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 10.9T + 47T^{2} \)
53 \( 1 + (0.707 + 0.408i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 2.74T + 59T^{2} \)
61 \( 1 - 7.20iT - 61T^{2} \)
67 \( 1 - 11.6T + 67T^{2} \)
71 \( 1 + 10.4iT - 71T^{2} \)
73 \( 1 + (-13.6 - 7.88i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 12.3T + 79T^{2} \)
83 \( 1 + (-4.03 + 6.99i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.60 + 7.98i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.00 - 4.04i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.971158656392346112279679489123, −9.267366792010574585645055206479, −8.591437338051657560708664402498, −7.52642693503698423875995633992, −6.53289321159604868803359039904, −5.68994477376941302625544775544, −4.51425756694011681486008151542, −3.87801869565803572517473885197, −2.07339192129795538923763337217, −0.808743355037984123913951952481, 1.99416217820741435873762943225, 2.77194743382984158716872972663, 4.09406872761075614191894473102, 5.38650221693910814440312463860, 6.28385581368752684574100789127, 6.89146774371732486594376385172, 7.937441512286369682367952397790, 9.113147844253408029081682813714, 9.703047378361461537136127900272, 10.47183221916250490262470673326

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