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Properties

Label 2-756-28.3-c1-0-13
Degree $2$
Conductor $756$
Sign $0.929 - 0.368i$
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  + (−0.953 − 1.04i)2-s + (−0.183 + 1.99i)4-s + (0.703 − 0.406i)5-s + (−2.60 − 0.476i)7-s + (2.25 − 1.70i)8-s + (−1.09 − 0.348i)10-s + (1.86 + 1.07i)11-s + 5.14i·13-s + (1.98 + 3.17i)14-s + (−3.93 − 0.729i)16-s + (1.78 + 1.02i)17-s + (−1.57 − 2.72i)19-s + (0.680 + 1.47i)20-s + (−0.651 − 2.96i)22-s + (−1.64 + 0.947i)23-s + ⋯
L(s)  = 1  + (−0.673 − 0.738i)2-s + (−0.0915 + 0.995i)4-s + (0.314 − 0.181i)5-s + (−0.983 − 0.179i)7-s + (0.797 − 0.603i)8-s + (−0.346 − 0.110i)10-s + (0.561 + 0.324i)11-s + 1.42i·13-s + (0.530 + 0.847i)14-s + (−0.983 − 0.182i)16-s + (0.431 + 0.249i)17-s + (−0.361 − 0.625i)19-s + (0.152 + 0.330i)20-s + (−0.138 − 0.632i)22-s + (−0.342 + 0.197i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.368i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.929 - 0.368i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.929 - 0.368i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.889963 + 0.170129i\)
\(L(\frac12)\) \(\approx\) \(0.889963 + 0.170129i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.953 + 1.04i)T \)
3 \( 1 \)
7 \( 1 + (2.60 + 0.476i)T \)
good5 \( 1 + (-0.703 + 0.406i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.86 - 1.07i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.14iT - 13T^{2} \)
17 \( 1 + (-1.78 - 1.02i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.57 + 2.72i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.64 - 0.947i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.65T + 29T^{2} \)
31 \( 1 + (-0.513 + 0.889i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.94 - 5.10i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.55iT - 41T^{2} \)
43 \( 1 - 10.2iT - 43T^{2} \)
47 \( 1 + (-1.06 - 1.83i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.32 + 5.75i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.32 - 10.9i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.15 + 2.97i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.91 - 3.99i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 11.0iT - 71T^{2} \)
73 \( 1 + (-8.64 - 4.99i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.82 - 3.94i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 17.5T + 83T^{2} \)
89 \( 1 + (-0.484 + 0.279i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 1.84iT - 97T^{2} \)
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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

−10.15655412530531406349000013661, −9.562148641241942651702778627895, −9.049663747702241413462020283168, −8.009748793009675851689151592491, −6.89330402198690647835513583140, −6.32231974220059605065997287613, −4.63924729276262663083256127034, −3.76780417506027039868517325995, −2.58715351412379569107929461273, −1.30633270936039808308877049230, 0.64184235558196658827077030392, 2.45975198539751460070929287257, 3.79477950965326551278643623852, 5.31658254918651872041074414777, 6.06429205249995456736301622039, 6.69751265311608450108962808152, 7.79407685936167977749183989253, 8.533240211616110337936589083574, 9.427464142419741837429771019137, 10.23210204926958739341205648743

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