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Properties

Label 2-756-63.41-c3-0-1
Degree $2$
Conductor $756$
Sign $-0.710 + 0.703i$
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  + (−8.29 + 14.3i)5-s + (18.2 + 3.26i)7-s + (−46.2 + 26.7i)11-s + (11.0 + 6.37i)13-s − 96.7·17-s + 54.6i·19-s + (55.6 + 32.1i)23-s + (−75.2 − 130. i)25-s + (112. − 65.0i)29-s + (−190. − 110. i)31-s + (−198. + 234. i)35-s + 279.·37-s + (−185. + 320. i)41-s + (−153. − 266. i)43-s + (163. + 283. i)47-s + ⋯
L(s)  = 1  + (−0.742 + 1.28i)5-s + (0.984 + 0.176i)7-s + (−1.26 + 0.732i)11-s + (0.235 + 0.136i)13-s − 1.38·17-s + 0.660i·19-s + (0.504 + 0.291i)23-s + (−0.601 − 1.04i)25-s + (0.721 − 0.416i)29-s + (−1.10 − 0.637i)31-s + (−0.957 + 1.13i)35-s + 1.24·37-s + (−0.705 + 1.22i)41-s + (−0.545 − 0.945i)43-s + (0.507 + 0.879i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.710 + 0.703i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.710 + 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.710 + 0.703i$
Analytic conductor: \(44.6054\)
Root analytic conductor: \(6.67873\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :3/2),\ -0.710 + 0.703i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3132541705\)
\(L(\frac12)\) \(\approx\) \(0.3132541705\)
\(L(\frac{5}{2})\) 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 \( 1 + (-18.2 - 3.26i)T \)
good5 \( 1 + (8.29 - 14.3i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (46.2 - 26.7i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-11.0 - 6.37i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 96.7T + 4.91e3T^{2} \)
19 \( 1 - 54.6iT - 6.85e3T^{2} \)
23 \( 1 + (-55.6 - 32.1i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-112. + 65.0i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (190. + 110. i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 279.T + 5.06e4T^{2} \)
41 \( 1 + (185. - 320. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (153. + 266. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-163. - 283. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 451. iT - 1.48e5T^{2} \)
59 \( 1 + (-258. + 448. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (234. - 135. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (370. - 642. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 914. iT - 3.57e5T^{2} \)
73 \( 1 + 337. iT - 3.89e5T^{2} \)
79 \( 1 + (498. + 863. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-17.0 - 29.4i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 208.T + 7.04e5T^{2} \)
97 \( 1 + (1.10e3 - 635. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−10.69961166581209524189427079668, −9.810912768434113787941665242858, −8.558629848099641398248645179631, −7.77054555775178624261318076092, −7.21310185396257622703430732286, −6.20332973959373825748236145340, −4.98569891970577324167326036135, −4.12073994002584604837922086147, −2.87328766602346980332153643375, −1.95622894415281814215266301385, 0.090586550475400283783476910332, 1.15070543808620206066650792118, 2.62897531813518710198835087296, 4.10052692574218845691959688247, 4.86594337184334844878009727921, 5.49667858906422663189641039949, 6.97526291198925786876606357047, 7.920547124710294203575319894043, 8.579914356749745694021182521522, 9.006423576357034793820571860895

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