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Properties

Label 2-756-21.17-c3-0-6
Degree $2$
Conductor $756$
Sign $-0.999 - 0.0178i$
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  + (7.05 + 12.2i)5-s + (−2.66 + 18.3i)7-s + (−14.4 − 8.36i)11-s + 21.2i·13-s + (−38.7 + 67.1i)17-s + (−1.43 + 0.826i)19-s + (−35.6 + 20.5i)23-s + (−37.1 + 64.3i)25-s + 100. i·29-s + (−45.9 − 26.5i)31-s + (−242. + 96.7i)35-s + (−101. − 176. i)37-s + 347.·41-s − 113.·43-s + (−45.7 − 79.2i)47-s + ⋯
L(s)  = 1  + (0.631 + 1.09i)5-s + (−0.144 + 0.989i)7-s + (−0.396 − 0.229i)11-s + 0.454i·13-s + (−0.553 + 0.958i)17-s + (−0.0172 + 0.00998i)19-s + (−0.323 + 0.186i)23-s + (−0.297 + 0.514i)25-s + 0.645i·29-s + (−0.265 − 0.153i)31-s + (−1.17 + 0.467i)35-s + (−0.452 − 0.783i)37-s + 1.32·41-s − 0.402·43-s + (−0.141 − 0.245i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0178i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0178i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.999 - 0.0178i$
Analytic conductor: \(44.6054\)
Root analytic conductor: \(6.67873\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :3/2),\ -0.999 - 0.0178i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.148657309\)
\(L(\frac12)\) \(\approx\) \(1.148657309\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.66 - 18.3i)T \)
good5 \( 1 + (-7.05 - 12.2i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (14.4 + 8.36i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 21.2iT - 2.19e3T^{2} \)
17 \( 1 + (38.7 - 67.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (1.43 - 0.826i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (35.6 - 20.5i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 100. iT - 2.43e4T^{2} \)
31 \( 1 + (45.9 + 26.5i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (101. + 176. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 347.T + 6.89e4T^{2} \)
43 \( 1 + 113.T + 7.95e4T^{2} \)
47 \( 1 + (45.7 + 79.2i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (22.2 + 12.8i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-450. + 781. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (402. - 232. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (173. - 300. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 620. iT - 3.57e5T^{2} \)
73 \( 1 + (564. + 326. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-68.3 - 118. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 298.T + 5.71e5T^{2} \)
89 \( 1 + (-422. - 732. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 494. iT - 9.12e5T^{2} \)
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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

−10.48594159801605627177622292089, −9.505388907211888475850100770755, −8.794709508328777152217892924291, −7.78450428150014216383800607046, −6.66268612043090363912832847479, −6.10055508805653352924750728488, −5.20883398848806543763339386719, −3.77274553554503689916659467813, −2.66579071355549317393076311147, −1.88260850012461048774750058184, 0.29921776641251474261957497415, 1.38271176782227245668620498326, 2.75080373007396616856289703838, 4.18741632914210145586582958471, 4.93848078806656426013815156718, 5.85392510425481811039978014488, 6.96610005051950479969304929338, 7.81328005433913695146575989207, 8.749630144373137985364385897253, 9.584348712071983437168312660108

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