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Properties

Label 2-756-12.11-c1-0-40
Degree $2$
Conductor $756$
Sign $0.376 + 0.926i$
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  + (1.38 − 0.270i)2-s + (1.85 − 0.752i)4-s − 1.41i·5-s − i·7-s + (2.36 − 1.54i)8-s + (−0.383 − 1.96i)10-s + 0.0560·11-s − 1.75·13-s + (−0.270 − 1.38i)14-s + (2.86 − 2.78i)16-s − 7.23i·17-s + 2.28i·19-s + (−1.06 − 2.62i)20-s + (0.0778 − 0.0151i)22-s + 3.19·23-s + ⋯
L(s)  = 1  + (0.981 − 0.191i)2-s + (0.926 − 0.376i)4-s − 0.633i·5-s − 0.377i·7-s + (0.837 − 0.546i)8-s + (−0.121 − 0.621i)10-s + 0.0169·11-s − 0.487·13-s + (−0.0724 − 0.370i)14-s + (0.717 − 0.696i)16-s − 1.75i·17-s + 0.523i·19-s + (−0.238 − 0.586i)20-s + (0.0165 − 0.00324i)22-s + 0.666·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.376 + 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.376 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.376 + 0.926i$
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: \((2,\ 756,\ (\ :1/2),\ 0.376 + 0.926i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.33493 - 1.57218i\)
\(L(\frac12)\) \(\approx\) \(2.33493 - 1.57218i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.38 + 0.270i)T \)
3 \( 1 \)
7 \( 1 + iT \)
good5 \( 1 + 1.41iT - 5T^{2} \)
11 \( 1 - 0.0560T + 11T^{2} \)
13 \( 1 + 1.75T + 13T^{2} \)
17 \( 1 + 7.23iT - 17T^{2} \)
19 \( 1 - 2.28iT - 19T^{2} \)
23 \( 1 - 3.19T + 23T^{2} \)
29 \( 1 - 4.78iT - 29T^{2} \)
31 \( 1 - 6.33iT - 31T^{2} \)
37 \( 1 + 5.24T + 37T^{2} \)
41 \( 1 + 3.58iT - 41T^{2} \)
43 \( 1 - 3.12iT - 43T^{2} \)
47 \( 1 + 5.27T + 47T^{2} \)
53 \( 1 + 5.18iT - 53T^{2} \)
59 \( 1 - 3.27T + 59T^{2} \)
61 \( 1 - 8.82T + 61T^{2} \)
67 \( 1 - 15.8iT - 67T^{2} \)
71 \( 1 - 14.2T + 71T^{2} \)
73 \( 1 + 5.03T + 73T^{2} \)
79 \( 1 - 12.1iT - 79T^{2} \)
83 \( 1 - 1.28T + 83T^{2} \)
89 \( 1 + 6.02iT - 89T^{2} \)
97 \( 1 + 16.4T + 97T^{2} \)
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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

−10.31327799621850145154246139365, −9.488083565261900961104180193393, −8.457680983649343085869013822034, −7.21979647508985229680274687354, −6.76609645114889170304793773691, −5.19476943801723973180525832745, −5.01051819065299501514208878422, −3.71484777541966076564232752606, −2.66264425561340356546086926037, −1.14291884730106713122047031918, 2.01572381874869603304199362319, 3.05427936305385798733595566949, 4.08326811126298331637151418593, 5.13182521716713530716311530298, 6.11750705792380889965305317883, 6.77278686736704994599520830562, 7.73755567143319569940086233706, 8.590608915145495181910846050291, 9.849602054508688590146214114388, 10.76342860009832289921723340194

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