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Properties

Label 2-756-252.187-c1-0-25
Degree $2$
Conductor $756$
Sign $-0.110 + 0.993i$
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  + (0.366 − 1.36i)2-s + (−1.73 − i)4-s + 3.46i·5-s + (−1.73 − 2i)7-s + (−2 + 1.99i)8-s + (4.73 + 1.26i)10-s − 2i·11-s + (4.5 − 2.59i)13-s + (−3.36 + 1.63i)14-s + (1.99 + 3.46i)16-s + (4.5 − 2.59i)17-s + (0.866 − 1.5i)19-s + (3.46 − 5.99i)20-s + (−2.73 − 0.732i)22-s − 4i·23-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)2-s + (−0.866 − 0.5i)4-s + 1.54i·5-s + (−0.654 − 0.755i)7-s + (−0.707 + 0.707i)8-s + (1.49 + 0.400i)10-s − 0.603i·11-s + (1.24 − 0.720i)13-s + (−0.899 + 0.436i)14-s + (0.499 + 0.866i)16-s + (1.09 − 0.630i)17-s + (0.198 − 0.344i)19-s + (0.774 − 1.34i)20-s + (−0.582 − 0.156i)22-s − 0.834i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.110 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.110 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.110 + 0.993i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.110 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.972401 - 1.08697i\)
\(L(\frac12)\) \(\approx\) \(0.972401 - 1.08697i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.366 + 1.36i)T \)
3 \( 1 \)
7 \( 1 + (1.73 + 2i)T \)
good5 \( 1 - 3.46iT - 5T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
13 \( 1 + (-4.5 + 2.59i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-4.5 + 2.59i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.866 + 1.5i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 + (-2.5 + 4.33i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.59 + 4.5i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.5 - 0.866i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-9.52 - 5.5i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.866 + 1.5i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.866 + 1.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.5 + 2.59i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.79 + 4.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 2iT - 71T^{2} \)
73 \( 1 + (10.5 - 6.06i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.59 - 1.5i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.59 - 4.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (7.5 + 4.33i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.5 - 0.866i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.20667378898954089370349598411, −9.792757842196541859419650441001, −8.489834726115911327578670104367, −7.51820100036712704374230749120, −6.39089047625110885524600654408, −5.79335277757109798687271734100, −4.21861343367615927474761168044, −3.24571085630085524185022414887, −2.80074448180339959554813795718, −0.807705777340967488330338476583, 1.35097707932548538516912250253, 3.44847187935673009890370698781, 4.37394542024994698816619912694, 5.43854291336814942701583897703, 5.91894941497485637499224891397, 7.04633598396296335277222799899, 8.133240245157347540838288542299, 8.841423112454200406979148225501, 9.260613588736545899517572758420, 10.24472171656277014905477703832

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