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Properties

Label 2-756-12.11-c1-0-42
Degree $2$
Conductor $756$
Sign $-0.653 + 0.757i$
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  + (−0.492 + 1.32i)2-s + (−1.51 − 1.30i)4-s − 3.62i·5-s − i·7-s + (2.47 − 1.36i)8-s + (4.81 + 1.78i)10-s − 0.830·11-s − 4.86·13-s + (1.32 + 0.492i)14-s + (0.586 + 3.95i)16-s − 3.28i·17-s + 6.45i·19-s + (−4.74 + 5.49i)20-s + (0.409 − 1.10i)22-s − 5.08·23-s + ⋯
L(s)  = 1  + (−0.348 + 0.937i)2-s + (−0.757 − 0.653i)4-s − 1.62i·5-s − 0.377i·7-s + (0.876 − 0.482i)8-s + (1.52 + 0.565i)10-s − 0.250·11-s − 1.35·13-s + (0.354 + 0.131i)14-s + (0.146 + 0.989i)16-s − 0.796i·17-s + 1.48i·19-s + (−1.06 + 1.22i)20-s + (0.0872 − 0.234i)22-s − 1.06·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.653 + 0.757i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.653 + 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.653 + 0.757i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.653 + 0.757i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.172463 - 0.376561i\)
\(L(\frac12)\) \(\approx\) \(0.172463 - 0.376561i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.492 - 1.32i)T \)
3 \( 1 \)
7 \( 1 + iT \)
good5 \( 1 + 3.62iT - 5T^{2} \)
11 \( 1 + 0.830T + 11T^{2} \)
13 \( 1 + 4.86T + 13T^{2} \)
17 \( 1 + 3.28iT - 17T^{2} \)
19 \( 1 - 6.45iT - 19T^{2} \)
23 \( 1 + 5.08T + 23T^{2} \)
29 \( 1 + 2.02iT - 29T^{2} \)
31 \( 1 - 7.12iT - 31T^{2} \)
37 \( 1 + 4.68T + 37T^{2} \)
41 \( 1 + 2.95iT - 41T^{2} \)
43 \( 1 + 1.62iT - 43T^{2} \)
47 \( 1 + 6.41T + 47T^{2} \)
53 \( 1 + 4.22iT - 53T^{2} \)
59 \( 1 - 8.31T + 59T^{2} \)
61 \( 1 + 5.55T + 61T^{2} \)
67 \( 1 + 11.6iT - 67T^{2} \)
71 \( 1 + 7.15T + 71T^{2} \)
73 \( 1 - 13.9T + 73T^{2} \)
79 \( 1 + 10.0iT - 79T^{2} \)
83 \( 1 - 16.5T + 83T^{2} \)
89 \( 1 - 15.9iT - 89T^{2} \)
97 \( 1 - 8.59T + 97T^{2} \)
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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

−9.779312217030234666535477211441, −9.120252823867099711108414838912, −8.144998543155372280017193678307, −7.70600251083830797024372932798, −6.58595195146178061477155141024, −5.35034620248731509332777648414, −4.95429231501718472436751455162, −3.91128429144591205450805557051, −1.70166751920724553448324790548, −0.22889851830755443512402684049, 2.21635879895454222142971779082, 2.79111141367310707582154023850, 3.91518850073388062822822434020, 5.11834843272155307464594035488, 6.42995416233729523188766625985, 7.34500524332865687739317110068, 8.073793436978934694873392000706, 9.259333347237088928408470024179, 10.00446541898312368963398470994, 10.60240168397235127436621458166

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