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Properties

Label 2-756-63.47-c1-0-7
Degree $2$
Conductor $756$
Sign $-0.949 + 0.313i$
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  − 0.0764·5-s + (−2.39 + 1.11i)7-s − 5.38i·11-s + (−4.60 + 2.65i)13-s + (1.89 + 3.27i)17-s + (−4.33 − 2.50i)19-s − 2.33i·23-s − 4.99·25-s + (−8.84 − 5.10i)29-s + (4.97 + 2.87i)31-s + (0.183 − 0.0852i)35-s + (0.354 − 0.613i)37-s + (−3.29 − 5.71i)41-s + (0.716 − 1.24i)43-s + (1.46 + 2.53i)47-s + ⋯
L(s)  = 1  − 0.0341·5-s + (−0.906 + 0.421i)7-s − 1.62i·11-s + (−1.27 + 0.737i)13-s + (0.458 + 0.794i)17-s + (−0.995 − 0.574i)19-s − 0.487i·23-s − 0.998·25-s + (−1.64 − 0.948i)29-s + (0.893 + 0.516i)31-s + (0.0309 − 0.0144i)35-s + (0.0582 − 0.100i)37-s + (−0.515 − 0.892i)41-s + (0.109 − 0.189i)43-s + (0.213 + 0.369i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 + 0.313i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.949 + 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.949 + 0.313i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.949 + 0.313i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0338732 - 0.210868i\)
\(L(\frac12)\) \(\approx\) \(0.0338732 - 0.210868i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.39 - 1.11i)T \)
good5 \( 1 + 0.0764T + 5T^{2} \)
11 \( 1 + 5.38iT - 11T^{2} \)
13 \( 1 + (4.60 - 2.65i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.89 - 3.27i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.33 + 2.50i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 2.33iT - 23T^{2} \)
29 \( 1 + (8.84 + 5.10i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.97 - 2.87i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.354 + 0.613i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.29 + 5.71i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.716 + 1.24i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.46 - 2.53i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (10.4 - 6.05i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.289 + 0.502i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.40 - 1.38i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.63 - 4.56i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.32iT - 71T^{2} \)
73 \( 1 + (6.17 - 3.56i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.469 + 0.812i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.49 + 11.2i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.51 + 2.62i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.18 - 3.56i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.902543898978220175758522895638, −9.105950285279937486694707584178, −8.378891973795308165181698397157, −7.33066159733243403385836931145, −6.26231500339414381105314533652, −5.74080020509493675725193651005, −4.36597928277504548519663325938, −3.31771482354761330939715702031, −2.21566877889240451109864745344, −0.099463409040947282969619068670, 1.99837098867106625201335384740, 3.23263803997700436477707570909, 4.37629268241043172276664160513, 5.30475535494057092341744167068, 6.46690513535952827178779509394, 7.37777502261085924098434680987, 7.84490990345856251021957517957, 9.436875070770372442323521684194, 9.774178417788118532457781947388, 10.45050910632011521673038131527

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