Properties

Label 2-756-63.47-c1-0-0
Degree $2$
Conductor $756$
Sign $-0.919 - 0.393i$
Analytic cond. $6.03669$
Root an. cond. $2.45696$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.74·5-s + (1.70 + 2.02i)7-s + 0.418i·11-s + (−1.32 + 0.765i)13-s + (−1.95 − 3.38i)17-s + (−5.11 − 2.95i)19-s + 8.92i·23-s + 2.52·25-s + (−6.00 − 3.46i)29-s + (−3.05 − 1.76i)31-s + (−4.67 − 5.55i)35-s + (−4.54 + 7.87i)37-s + (1.06 + 1.84i)41-s + (−5.77 + 10.0i)43-s + (0.885 + 1.53i)47-s + ⋯
L(s)  = 1  − 1.22·5-s + (0.644 + 0.764i)7-s + 0.126i·11-s + (−0.367 + 0.212i)13-s + (−0.473 − 0.820i)17-s + (−1.17 − 0.678i)19-s + 1.86i·23-s + 0.505·25-s + (−1.11 − 0.643i)29-s + (−0.548 − 0.316i)31-s + (−0.790 − 0.938i)35-s + (−0.747 + 1.29i)37-s + (0.165 + 0.287i)41-s + (−0.881 + 1.52i)43-s + (0.129 + 0.223i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.919 - 0.393i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.919 - 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.919 - 0.393i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.919 - 0.393i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0769633 + 0.374988i\)
\(L(\frac12)\) \(\approx\) \(0.0769633 + 0.374988i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.70 - 2.02i)T \)
good5 \( 1 + 2.74T + 5T^{2} \)
11 \( 1 - 0.418iT - 11T^{2} \)
13 \( 1 + (1.32 - 0.765i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.95 + 3.38i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.11 + 2.95i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 8.92iT - 23T^{2} \)
29 \( 1 + (6.00 + 3.46i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.05 + 1.76i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.54 - 7.87i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.06 - 1.84i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.77 - 10.0i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.885 - 1.53i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.39 - 1.96i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.02 + 3.51i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.61 - 0.932i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.38 + 11.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.51iT - 71T^{2} \)
73 \( 1 + (-1.65 + 0.952i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.433 - 0.751i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.45 + 5.99i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.88 - 8.46i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.200 + 0.115i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.19060904882190786262472140945, −9.678162746002040236704751269145, −8.993180205784353430688342165965, −8.010490730018295630508770868271, −7.49079245777757521797916738910, −6.40924051998158788516099999284, −5.15059890149594082508094380682, −4.42684710844381742511743857538, −3.27647037187494549741319405043, −1.94629896074510304494899420411, 0.18643313208070900878616013519, 2.02564511928053227223895570396, 3.76192265165939203057006372938, 4.19370238582229635164400445341, 5.35230863317543422541862004899, 6.69913966750387357614267766411, 7.41259398551853847062364225847, 8.310005227032754538477047111482, 8.762384421781234894875972057963, 10.38377288295912945046550158201

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