Properties

Label 2-756-63.4-c1-0-2
Degree $2$
Conductor $756$
Sign $0.972 - 0.234i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.90·5-s + (−2.43 + 1.03i)7-s + 3.06·11-s + (1.13 − 1.96i)13-s + (0.713 − 1.23i)17-s + (2.98 + 5.16i)19-s + 7.15·23-s − 1.37·25-s + (−0.468 − 0.810i)29-s + (4.11 + 7.11i)31-s + (−4.63 + 1.97i)35-s + (−1.41 − 2.45i)37-s + (5.31 − 9.20i)41-s + (2.98 + 5.16i)43-s + (−0.483 + 0.837i)47-s + ⋯
L(s)  = 1  + 0.851·5-s + (−0.920 + 0.391i)7-s + 0.924·11-s + (0.313 − 0.543i)13-s + (0.173 − 0.299i)17-s + (0.684 + 1.18i)19-s + 1.49·23-s − 0.275·25-s + (−0.0869 − 0.150i)29-s + (0.738 + 1.27i)31-s + (−0.782 + 0.333i)35-s + (−0.232 − 0.403i)37-s + (0.830 − 1.43i)41-s + (0.455 + 0.788i)43-s + (−0.0705 + 0.122i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.234i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 - 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.972 - 0.234i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.972 - 0.234i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73783 + 0.206756i\)
\(L(\frac12)\) \(\approx\) \(1.73783 + 0.206756i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.43 - 1.03i)T \)
good5 \( 1 - 1.90T + 5T^{2} \)
11 \( 1 - 3.06T + 11T^{2} \)
13 \( 1 + (-1.13 + 1.96i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.713 + 1.23i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.98 - 5.16i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 7.15T + 23T^{2} \)
29 \( 1 + (0.468 + 0.810i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.11 - 7.11i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.41 + 2.45i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.31 + 9.20i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.98 - 5.16i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.483 - 0.837i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.45 - 9.44i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.68 + 9.84i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.449 - 0.778i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.813 + 1.40i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.36T + 71T^{2} \)
73 \( 1 + (0.996 - 1.72i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.16 + 7.22i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.98 - 13.8i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.58 - 4.48i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.922 - 1.59i)T + (-48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.24083937624759235247026465838, −9.429714640560890457646119251639, −9.001348427173066376290815304846, −7.76654457607722545228649467600, −6.68878827969493266708429020137, −5.99283922778990466824216122433, −5.20594630303706090913277689758, −3.71772242504210065178842598563, −2.79724931865543122832227970783, −1.29676658614499144033608129306, 1.13775168616962812750866719739, 2.66046479425599718550152643975, 3.76863678775406306213962442683, 4.89938006564437742721685944840, 6.13405184965706786118258766777, 6.62122482147832434834557797246, 7.56978644817600626256805904219, 9.007001426757031445594979988424, 9.373119688833455535852552761427, 10.14213576416005893322176770369

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