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Properties

Label 2-756-36.23-c1-0-15
Degree $2$
Conductor $756$
Sign $0.456 - 0.889i$
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  + (−0.802 + 1.16i)2-s + (−0.712 − 1.86i)4-s + (2.73 + 1.57i)5-s + (0.866 − 0.5i)7-s + (2.74 + 0.670i)8-s + (−4.02 + 1.91i)10-s + (2.55 + 4.42i)11-s + (2.38 − 4.12i)13-s + (−0.112 + 1.40i)14-s + (−2.98 + 2.66i)16-s − 7.22i·17-s + 0.531i·19-s + (1.00 − 6.22i)20-s + (−7.20 − 0.576i)22-s + (−1.59 + 2.76i)23-s + ⋯
L(s)  = 1  + (−0.567 + 0.823i)2-s + (−0.356 − 0.934i)4-s + (1.22 + 0.705i)5-s + (0.327 − 0.188i)7-s + (0.971 + 0.237i)8-s + (−1.27 + 0.605i)10-s + (0.770 + 1.33i)11-s + (0.661 − 1.14i)13-s + (−0.0301 + 0.376i)14-s + (−0.746 + 0.665i)16-s − 1.75i·17-s + 0.121i·19-s + (0.224 − 1.39i)20-s + (−1.53 − 0.122i)22-s + (−0.332 + 0.576i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.456 - 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.456 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.456 - 0.889i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.456 - 0.889i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30795 + 0.799192i\)
\(L(\frac12)\) \(\approx\) \(1.30795 + 0.799192i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.802 - 1.16i)T \)
3 \( 1 \)
7 \( 1 + (-0.866 + 0.5i)T \)
good5 \( 1 + (-2.73 - 1.57i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.55 - 4.42i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.38 + 4.12i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 7.22iT - 17T^{2} \)
19 \( 1 - 0.531iT - 19T^{2} \)
23 \( 1 + (1.59 - 2.76i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.50 + 0.871i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.47 + 0.851i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 3.11T + 37T^{2} \)
41 \( 1 + (-2.32 - 1.34i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.39 + 2.53i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.47 + 2.55i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 13.6iT - 53T^{2} \)
59 \( 1 + (3.71 - 6.42i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.23 - 5.60i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.72 + 0.997i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.0T + 71T^{2} \)
73 \( 1 + 6.59T + 73T^{2} \)
79 \( 1 + (-1.73 + 0.998i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.28 - 5.68i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 9.77iT - 89T^{2} \)
97 \( 1 + (6.15 + 10.6i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.19540563265427996395294732156, −9.624761653481839469021913715493, −8.958307908499965162513447262785, −7.63171373636521159707280159069, −7.13426640541150234961082835475, −6.14945700886667751633289132114, −5.46118225584591403972008773267, −4.36922591483557274897014164317, −2.60831015285668650377092802319, −1.32439726174435529527545087712, 1.24756771996209201292235344125, 2.02371562839134831695179701798, 3.56609851631415179197121162076, 4.52019471418197137386854835761, 5.86635873173085181456902697120, 6.50529113410622903599451504450, 8.137264246158982493102349362237, 8.769628871221969022295163838628, 9.206634192880525069189031050945, 10.18016628044627389395437919731

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