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Properties

Label 2-756-63.20-c3-0-13
Degree $2$
Conductor $756$
Sign $0.972 - 0.231i$
Analytic cond. $44.6054$
Root an. cond. $6.67873$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.99 + 5.19i)5-s + (16.2 − 8.93i)7-s + (39.3 + 22.7i)11-s + (22.7 − 13.1i)13-s − 19.7·17-s + 27.9i·19-s + (60.3 − 34.8i)23-s + (44.5 − 77.0i)25-s + (119. + 68.7i)29-s + (−138. + 79.7i)31-s + (95.0 + 57.4i)35-s − 287.·37-s + (−20.4 − 35.3i)41-s + (55.4 − 95.9i)43-s + (109. − 189. i)47-s + ⋯
L(s)  = 1  + (0.268 + 0.464i)5-s + (0.875 − 0.482i)7-s + (1.07 + 0.623i)11-s + (0.485 − 0.280i)13-s − 0.281·17-s + 0.337i·19-s + (0.547 − 0.315i)23-s + (0.356 − 0.616i)25-s + (0.762 + 0.440i)29-s + (−0.800 + 0.462i)31-s + (0.459 + 0.277i)35-s − 1.27·37-s + (−0.0778 − 0.134i)41-s + (0.196 − 0.340i)43-s + (0.340 − 0.589i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.231i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.972 - 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.972 - 0.231i$
Analytic conductor: \(44.6054\)
Root analytic conductor: \(6.67873\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (629, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :3/2),\ 0.972 - 0.231i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.705940062\)
\(L(\frac12)\) \(\approx\) \(2.705940062\)
\(L(\frac{5}{2})\) 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 \( 1 + (-16.2 + 8.93i)T \)
good5 \( 1 + (-2.99 - 5.19i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-39.3 - 22.7i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-22.7 + 13.1i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 19.7T + 4.91e3T^{2} \)
19 \( 1 - 27.9iT - 6.85e3T^{2} \)
23 \( 1 + (-60.3 + 34.8i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-119. - 68.7i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (138. - 79.7i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 287.T + 5.06e4T^{2} \)
41 \( 1 + (20.4 + 35.3i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-55.4 + 95.9i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-109. + 189. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 209. iT - 1.48e5T^{2} \)
59 \( 1 + (-413. - 716. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-594. - 343. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (171. + 296. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 387. iT - 3.57e5T^{2} \)
73 \( 1 + 220. iT - 3.89e5T^{2} \)
79 \( 1 + (-242. + 419. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (354. - 613. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 140.T + 7.04e5T^{2} \)
97 \( 1 + (-1.30e3 - 753. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−10.25102563288395711700843265925, −8.997520213280392618984916268931, −8.396147479942240374725680398155, −7.16251506064918151093989251726, −6.70727676494127589555140851513, −5.48653085328426114306313942697, −4.47458981783303895692253325880, −3.54618552565564691584776827818, −2.11799800902041710934509504635, −1.04150175444705977236831091408, 0.970876347265563139004004754146, 1.95516197872806198710807940258, 3.42175235282091408235099798779, 4.53995860306912283268367471365, 5.42569471050860229098301336131, 6.31623074895393118130943860312, 7.31233078512765207775176722640, 8.544577098363453455987167145741, 8.860142698830806734662950160991, 9.736305896540270319910561663472

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