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Properties

Label 2-756-252.95-c1-0-9
Degree $2$
Conductor $756$
Sign $-0.355 - 0.934i$
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  + (−1.37 + 0.320i)2-s + (1.79 − 0.883i)4-s + 1.64i·5-s + (−1.79 + 1.94i)7-s + (−2.18 + 1.79i)8-s + (−0.526 − 2.26i)10-s + 6.12·11-s + (0.599 − 1.03i)13-s + (1.85 − 3.25i)14-s + (2.43 − 3.17i)16-s + (2.89 + 1.66i)17-s + (−3.61 + 2.08i)19-s + (1.45 + 2.94i)20-s + (−8.43 + 1.96i)22-s − 8.43·23-s + ⋯
L(s)  = 1  + (−0.973 + 0.226i)2-s + (0.897 − 0.441i)4-s + 0.734i·5-s + (−0.679 + 0.733i)7-s + (−0.773 + 0.633i)8-s + (−0.166 − 0.715i)10-s + 1.84·11-s + (0.166 − 0.288i)13-s + (0.495 − 0.868i)14-s + (0.609 − 0.792i)16-s + (0.701 + 0.404i)17-s + (−0.829 + 0.478i)19-s + (0.324 + 0.658i)20-s + (−1.79 + 0.418i)22-s − 1.75·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.355 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.355 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.355 - 0.934i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.355 - 0.934i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.480067 + 0.696360i\)
\(L(\frac12)\) \(\approx\) \(0.480067 + 0.696360i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.37 - 0.320i)T \)
3 \( 1 \)
7 \( 1 + (1.79 - 1.94i)T \)
good5 \( 1 - 1.64iT - 5T^{2} \)
11 \( 1 - 6.12T + 11T^{2} \)
13 \( 1 + (-0.599 + 1.03i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.89 - 1.66i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.61 - 2.08i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 8.43T + 23T^{2} \)
29 \( 1 + (-0.398 + 0.230i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.20 - 2.42i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.65 - 4.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.25 - 2.45i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.733 + 0.423i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.21 - 7.30i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.122 + 0.0707i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.42 - 9.39i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.10 - 1.91i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.71 - 2.14i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.07T + 71T^{2} \)
73 \( 1 + (5.37 - 9.31i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.67 - 3.85i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.20 + 3.81i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (8.60 - 4.97i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.92 - 6.79i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.32963639391545438150414997710, −9.755985265941398041595075268971, −8.918744652787772801562326156134, −8.192480334261777164429995540953, −7.08998747424551903412513866034, −6.24692161671492876416163606811, −5.90149969141601652527741246574, −3.94982749621312865878482960137, −2.86083358112041542208225656717, −1.54374474099506046766962128102, 0.61836652456214423583501051003, 1.85653792114883089503210705674, 3.54345430151242819263055095494, 4.26771785362575981198613823801, 6.00723600161839617480065981651, 6.70464378584400378427258874040, 7.57139793527268302540146993381, 8.619237410070161798359530957251, 9.285500094265149405237185076980, 9.840135722022071198149866707261

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