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Properties

Label 2-756-63.41-c3-0-18
Degree $2$
Conductor $756$
Sign $-0.387 + 0.921i$
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 + (2.10 − 18.4i)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 + (0.113 − 0.993i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.387 + 0.921i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.387 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.387 + 0.921i$
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: \((2,\ 756,\ (\ :3/2),\ -0.387 + 0.921i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7707199867\)
\(L(\frac12)\) \(\approx\) \(0.7707199867\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.10 + 18.4i)T \)
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

−9.976048183474870184333828859278, −8.734751134767219122733089033805, −7.74488011796106666865183444918, −7.17744701863286584178685210310, −6.34828901161083655865277690449, −5.13169050299949793798031399914, −3.88456728047021395325169454289, −3.32274757142229455164013602079, −1.78015894674925920251453814802, −0.22268137180555881676779842019, 1.21486867245639750720112446458, 2.56780934039110691613077379179, 3.83982398204934621390728667689, 4.90441329838927975385322997108, 5.60224296360731881591596374503, 6.69599258351693931974095095866, 7.923286086514336488331673039654, 8.454944144350816208219351358814, 9.260908680498562848861724801991, 10.04176081562368178568254411332

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