Properties

Label 2-756-63.20-c3-0-0
Degree $2$
Conductor $756$
Sign $-0.768 + 0.639i$
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  + (6.03 + 10.4i)5-s + (−16.9 + 7.37i)7-s + (−0.00221 − 0.00127i)11-s + (−6.06 + 3.50i)13-s − 28.3·17-s + 49.1i·19-s + (−44.4 + 25.6i)23-s + (−10.3 + 17.9i)25-s + (−97.9 − 56.5i)29-s + (28.4 − 16.4i)31-s + (−179. − 133. i)35-s − 101.·37-s + (−11.2 − 19.4i)41-s + (−227. + 393. i)43-s + (231. − 400. i)47-s + ⋯
L(s)  = 1  + (0.539 + 0.935i)5-s + (−0.917 + 0.398i)7-s + (−6.06e−5 − 3.50e−5i)11-s + (−0.129 + 0.0747i)13-s − 0.403·17-s + 0.594i·19-s + (−0.403 + 0.232i)23-s + (−0.0828 + 0.143i)25-s + (−0.627 − 0.362i)29-s + (0.164 − 0.0950i)31-s + (−0.867 − 0.642i)35-s − 0.453·37-s + (−0.0428 − 0.0741i)41-s + (−0.805 + 1.39i)43-s + (0.717 − 1.24i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.768 + 0.639i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.768 + 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.768 + 0.639i$
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: \((2,\ 756,\ (\ :3/2),\ -0.768 + 0.639i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.07611675859\)
\(L(\frac12)\) \(\approx\) \(0.07611675859\)
\(L(\frac{5}{2})\) 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 \( 1 + (16.9 - 7.37i)T \)
good5 \( 1 + (-6.03 - 10.4i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (0.00221 + 0.00127i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (6.06 - 3.50i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 28.3T + 4.91e3T^{2} \)
19 \( 1 - 49.1iT - 6.85e3T^{2} \)
23 \( 1 + (44.4 - 25.6i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (97.9 + 56.5i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-28.4 + 16.4i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 101.T + 5.06e4T^{2} \)
41 \( 1 + (11.2 + 19.4i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (227. - 393. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-231. + 400. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 567. iT - 1.48e5T^{2} \)
59 \( 1 + (145. + 252. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (592. + 341. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-269. - 467. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 307. iT - 3.57e5T^{2} \)
73 \( 1 + 495. iT - 3.89e5T^{2} \)
79 \( 1 + (324. - 562. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-565. + 979. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 130.T + 7.04e5T^{2} \)
97 \( 1 + (1.26e3 + 727. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−10.24825729065930251559063159344, −9.808481100168437246262089882521, −8.903606409811124566592189926232, −7.83364037950148159549301485073, −6.74965963536324203369877384020, −6.25911749601165860683550813973, −5.30280428684611885648383464516, −3.85801819310913329347740499520, −2.90793580417419014773307574568, −1.93200329915494163804858392877, 0.02021602692386872212721342457, 1.28896550544489100506054488491, 2.66090158839603231423452219748, 3.90897025240174314889853442766, 4.91568002380333139639209322263, 5.84300857414880402008735564524, 6.74740005725713461083654824423, 7.66251599456820334048572273675, 8.903332993070372161206800995310, 9.256068102366352417369558448322

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