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Properties

Label 2-756-63.20-c3-0-23
Degree $2$
Conductor $756$
Sign $-0.600 - 0.799i$
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  + (−7.82 − 13.5i)5-s + (−13.7 − 12.4i)7-s + (−34.2 − 19.7i)11-s + (55.5 − 32.0i)13-s − 56.6·17-s − 117. i·19-s + (6.59 − 3.80i)23-s + (−59.9 + 103. i)25-s + (−39.8 − 22.9i)29-s + (251. − 145. i)31-s + (−60.6 + 283. i)35-s − 335.·37-s + (−97.2 − 168. i)41-s + (152. − 263. i)43-s + (−318. + 550. i)47-s + ⋯
L(s)  = 1  + (−0.699 − 1.21i)5-s + (−0.742 − 0.670i)7-s + (−0.937 − 0.541i)11-s + (1.18 − 0.684i)13-s − 0.808·17-s − 1.41i·19-s + (0.0597 − 0.0345i)23-s + (−0.479 + 0.830i)25-s + (−0.254 − 0.147i)29-s + (1.45 − 0.842i)31-s + (−0.293 + 1.36i)35-s − 1.49·37-s + (−0.370 − 0.641i)41-s + (0.540 − 0.935i)43-s + (−0.987 + 1.70i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.600 - 0.799i$
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: \((2,\ 756,\ (\ :3/2),\ -0.600 - 0.799i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5677977023\)
\(L(\frac12)\) \(\approx\) \(0.5677977023\)
\(L(\frac{5}{2})\) 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 \( 1 + (13.7 + 12.4i)T \)
good5 \( 1 + (7.82 + 13.5i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (34.2 + 19.7i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-55.5 + 32.0i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 56.6T + 4.91e3T^{2} \)
19 \( 1 + 117. iT - 6.85e3T^{2} \)
23 \( 1 + (-6.59 + 3.80i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (39.8 + 22.9i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-251. + 145. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 335.T + 5.06e4T^{2} \)
41 \( 1 + (97.2 + 168. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-152. + 263. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (318. - 550. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 274. iT - 1.48e5T^{2} \)
59 \( 1 + (-258. - 448. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-142. - 82.1i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-368. - 637. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 599. iT - 3.57e5T^{2} \)
73 \( 1 + 214. iT - 3.89e5T^{2} \)
79 \( 1 + (454. - 786. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-389. + 675. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 443.T + 7.04e5T^{2} \)
97 \( 1 + (-337. - 194. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−9.180968013692065940799788554435, −8.555115971180049667723125171153, −7.83846366827575245884281222053, −6.80434762371397653228604340804, −5.75455916721503532134050673027, −4.74334371041953288884944901893, −3.89679356332762602453165030788, −2.80961448033111888952289307636, −0.907259096284257041663101757373, −0.19500713610379116321286214375, 1.96138647483962563010700988346, 3.11630847721928125272176875376, 3.82498317104767017635153464118, 5.18946908494086666115939956093, 6.45494058066688508119437017806, 6.77020058542970026459503413608, 7.987882961056927478126360356558, 8.648211423280024527479425543100, 9.859886748116754630117642521995, 10.46062192860396395440668560055

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