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Properties

Label 2-756-63.20-c3-0-12
Degree $2$
Conductor $756$
Sign $0.457 + 0.889i$
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  + (−9.12 − 15.7i)5-s + (−9.48 + 15.9i)7-s + (49.3 + 28.4i)11-s + (9.36 − 5.40i)13-s − 65.2·17-s − 36.6i·19-s + (70.2 − 40.5i)23-s + (−103. + 179. i)25-s + (233. + 134. i)29-s + (−117. + 67.9i)31-s + (337. + 4.68i)35-s − 125.·37-s + (117. + 203. i)41-s + (22.6 − 39.2i)43-s + (241. − 417. i)47-s + ⋯
L(s)  = 1  + (−0.815 − 1.41i)5-s + (−0.511 + 0.859i)7-s + (1.35 + 0.780i)11-s + (0.199 − 0.115i)13-s − 0.931·17-s − 0.441i·19-s + (0.636 − 0.367i)23-s + (−0.830 + 1.43i)25-s + (1.49 + 0.863i)29-s + (−0.682 + 0.393i)31-s + (1.63 + 0.0226i)35-s − 0.558·37-s + (0.447 + 0.775i)41-s + (0.0802 − 0.139i)43-s + (0.748 − 1.29i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.457 + 0.889i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.457 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.457 + 0.889i$
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: \((2,\ 756,\ (\ :3/2),\ 0.457 + 0.889i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.464571635\)
\(L(\frac12)\) \(\approx\) \(1.464571635\)
\(L(\frac{5}{2})\) 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 \( 1 + (9.48 - 15.9i)T \)
good5 \( 1 + (9.12 + 15.7i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-49.3 - 28.4i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-9.36 + 5.40i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 65.2T + 4.91e3T^{2} \)
19 \( 1 + 36.6iT - 6.85e3T^{2} \)
23 \( 1 + (-70.2 + 40.5i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-233. - 134. i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (117. - 67.9i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 125.T + 5.06e4T^{2} \)
41 \( 1 + (-117. - 203. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-22.6 + 39.2i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-241. + 417. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 70.1iT - 1.48e5T^{2} \)
59 \( 1 + (176. + 306. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-512. - 295. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (261. + 453. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 895. iT - 3.57e5T^{2} \)
73 \( 1 + 982. iT - 3.89e5T^{2} \)
79 \( 1 + (-510. + 883. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (152. - 263. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 1.02e3T + 7.04e5T^{2} \)
97 \( 1 + (677. + 391. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−9.396505062935303381233960000874, −8.940263589532344870146257288084, −8.423345050371605075503007573664, −7.12554217112191925879068151984, −6.35674400971981334410467921652, −5.04909439973932036366499912120, −4.46421964465080399614926932788, −3.35824520387959619435449274987, −1.80707984534524228572343297268, −0.54559703859382216540239259833, 0.890872153309908023395921106632, 2.69841646632579656632822679023, 3.71438884083873781548624870366, 4.17708737888445982337828793971, 6.02505931137292042143843565753, 6.73815778880913528338363978703, 7.25664404037864658110486227625, 8.314727521156819305172326172275, 9.294700898142367985319323713334, 10.27023054533377002798231999839

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