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Properties

Label 2-756-84.23-c1-0-43
Degree $2$
Conductor $756$
Sign $0.999 - 0.00601i$
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  + (1.41 − 0.0220i)2-s + (1.99 − 0.0624i)4-s + (−0.303 − 0.175i)5-s + (2.63 + 0.234i)7-s + (2.82 − 0.132i)8-s + (−0.432 − 0.240i)10-s + (0.356 + 0.617i)11-s + 0.127·13-s + (3.73 + 0.272i)14-s + (3.99 − 0.249i)16-s + (−5.30 + 3.06i)17-s + (2.91 + 1.68i)19-s + (−0.617 − 0.331i)20-s + (0.517 + 0.864i)22-s + (2.38 − 4.13i)23-s + ⋯
L(s)  = 1  + (0.999 − 0.0156i)2-s + (0.999 − 0.0312i)4-s + (−0.135 − 0.0783i)5-s + (0.996 + 0.0884i)7-s + (0.998 − 0.0467i)8-s + (−0.136 − 0.0761i)10-s + (0.107 + 0.186i)11-s + 0.0353·13-s + (0.997 + 0.0729i)14-s + (0.998 − 0.0623i)16-s + (−1.28 + 0.742i)17-s + (0.669 + 0.386i)19-s + (−0.138 − 0.0740i)20-s + (0.110 + 0.184i)22-s + (0.497 − 0.862i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00601i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00601i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.999 - 0.00601i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.999 - 0.00601i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.13832 + 0.00943294i\)
\(L(\frac12)\) \(\approx\) \(3.13832 + 0.00943294i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 + 0.0220i)T \)
3 \( 1 \)
7 \( 1 + (-2.63 - 0.234i)T \)
good5 \( 1 + (0.303 + 0.175i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.356 - 0.617i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.127T + 13T^{2} \)
17 \( 1 + (5.30 - 3.06i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.91 - 1.68i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.38 + 4.13i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.39iT - 29T^{2} \)
31 \( 1 + (-0.00202 + 0.00116i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.17 + 3.76i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 8.22iT - 41T^{2} \)
43 \( 1 - 7.55iT - 43T^{2} \)
47 \( 1 + (-1.33 + 2.30i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (10.1 - 5.86i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.13 + 8.90i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.83 - 4.91i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.79 + 4.50i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.61T + 71T^{2} \)
73 \( 1 + (2.15 + 3.72i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.92 + 1.10i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.12T + 83T^{2} \)
89 \( 1 + (7.79 + 4.49i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 17.3T + 97T^{2} \)
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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

−10.73675745049751124380524599475, −9.605193645354433105372871231705, −8.353542146945057143159498913974, −7.74791784937048443241208790416, −6.62744999331283451671940339870, −5.85586657716824591663480234672, −4.67092897908348908643453531110, −4.21578539369350981151319433080, −2.75254659117128376375367140239, −1.61591838525667275374350797195, 1.56696754957566075070145997318, 2.85290186683325729280398039420, 3.99909670857303804912040861680, 4.94760943635239614276684118401, 5.61975970016080769333682697016, 6.94146405253861035048898198069, 7.41984115714931277248956576623, 8.521330711211525943980567625931, 9.513499805738555842734770423104, 10.80001017964390336759375651437

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