Properties

Label 2-756-108.11-c1-0-100
Degree $2$
Conductor $756$
Sign $-0.472 + 0.881i$
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  + (1.26 − 0.636i)2-s + (0.577 − 1.63i)3-s + (1.18 − 1.60i)4-s + (−0.161 + 0.442i)5-s + (−0.311 − 2.42i)6-s + (0.984 + 0.173i)7-s + (0.478 − 2.78i)8-s + (−2.33 − 1.88i)9-s + (0.0783 + 0.661i)10-s + (−2.51 + 0.914i)11-s + (−1.93 − 2.87i)12-s + (0.804 − 0.675i)13-s + (1.35 − 0.407i)14-s + (0.629 + 0.518i)15-s + (−1.17 − 3.82i)16-s + (5.28 − 3.05i)17-s + ⋯
L(s)  = 1  + (0.892 − 0.450i)2-s + (0.333 − 0.942i)3-s + (0.594 − 0.803i)4-s + (−0.0720 + 0.197i)5-s + (−0.126 − 0.991i)6-s + (0.372 + 0.0656i)7-s + (0.169 − 0.985i)8-s + (−0.778 − 0.628i)9-s + (0.0247 + 0.209i)10-s + (−0.757 + 0.275i)11-s + (−0.559 − 0.828i)12-s + (0.223 − 0.187i)13-s + (0.361 − 0.108i)14-s + (0.162 + 0.133i)15-s + (−0.292 − 0.956i)16-s + (1.28 − 0.739i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.472 + 0.881i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.472 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.472 + 0.881i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (659, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.472 + 0.881i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43396 - 2.39603i\)
\(L(\frac12)\) \(\approx\) \(1.43396 - 2.39603i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.26 + 0.636i)T \)
3 \( 1 + (-0.577 + 1.63i)T \)
7 \( 1 + (-0.984 - 0.173i)T \)
good5 \( 1 + (0.161 - 0.442i)T + (-3.83 - 3.21i)T^{2} \)
11 \( 1 + (2.51 - 0.914i)T + (8.42 - 7.07i)T^{2} \)
13 \( 1 + (-0.804 + 0.675i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-5.28 + 3.05i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.94 + 1.12i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.02 + 5.81i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-0.485 + 0.578i)T + (-5.03 - 28.5i)T^{2} \)
31 \( 1 + (5.63 - 0.992i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (-5.10 - 8.84i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.917 - 1.09i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (-4.06 - 11.1i)T + (-32.9 + 27.6i)T^{2} \)
47 \( 1 + (-0.0862 + 0.489i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 - 9.90iT - 53T^{2} \)
59 \( 1 + (-9.24 - 3.36i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-1.06 + 6.05i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-5.34 - 6.37i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (2.25 + 3.91i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-7.88 + 13.6i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.32 - 3.96i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (10.1 + 8.49i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (2.16 + 1.25i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.27 - 1.19i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.29953930473600036976828116535, −9.286347185761894673380102438548, −8.099694680960466404674327970630, −7.38341825772024253432131352321, −6.47753912029319828593746470390, −5.56515898916718409231836513197, −4.61080544414862217640337908313, −3.18509411878324761249375713594, −2.47906306801800154417359637048, −1.09997491454847187409806004979, 2.23772525848881932471282230374, 3.53233108755951260512584738406, 4.13487169686813095756813579434, 5.40790745483569205322456780306, 5.67408623659453039506436907083, 7.22144383975311511557329016115, 8.093210339278339753461369466352, 8.647077971007374702169855254882, 9.854178849813803806320311557391, 10.76537742955053628201091216635

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