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Properties

Label 2-756-12.11-c1-0-30
Degree $2$
Conductor $756$
Sign $0.875 - 0.483i$
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  + (1.21 + 0.718i)2-s + (0.967 + 1.75i)4-s − 1.77i·5-s − i·7-s + (−0.0793 + 2.82i)8-s + (1.27 − 2.16i)10-s + 5.69·11-s − 0.512·13-s + (0.718 − 1.21i)14-s + (−2.12 + 3.38i)16-s + 1.25i·17-s − 2.17i·19-s + (3.11 − 1.71i)20-s + (6.94 + 4.09i)22-s + 4.44·23-s + ⋯
L(s)  = 1  + (0.861 + 0.508i)2-s + (0.483 + 0.875i)4-s − 0.794i·5-s − 0.377i·7-s + (−0.0280 + 0.999i)8-s + (0.403 − 0.684i)10-s + 1.71·11-s − 0.142·13-s + (0.192 − 0.325i)14-s + (−0.532 + 0.846i)16-s + 0.304i·17-s − 0.499i·19-s + (0.695 − 0.384i)20-s + (1.48 + 0.873i)22-s + 0.926·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 - 0.483i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.875 - 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.875 - 0.483i$
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: \((2,\ 756,\ (\ :1/2),\ 0.875 - 0.483i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.64394 + 0.682005i\)
\(L(\frac12)\) \(\approx\) \(2.64394 + 0.682005i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.21 - 0.718i)T \)
3 \( 1 \)
7 \( 1 + iT \)
good5 \( 1 + 1.77iT - 5T^{2} \)
11 \( 1 - 5.69T + 11T^{2} \)
13 \( 1 + 0.512T + 13T^{2} \)
17 \( 1 - 1.25iT - 17T^{2} \)
19 \( 1 + 2.17iT - 19T^{2} \)
23 \( 1 - 4.44T + 23T^{2} \)
29 \( 1 - 4.23iT - 29T^{2} \)
31 \( 1 + 6.10iT - 31T^{2} \)
37 \( 1 + 2.77T + 37T^{2} \)
41 \( 1 - 9.84iT - 41T^{2} \)
43 \( 1 + 3.69iT - 43T^{2} \)
47 \( 1 - 3.24T + 47T^{2} \)
53 \( 1 + 2.68iT - 53T^{2} \)
59 \( 1 + 6.33T + 59T^{2} \)
61 \( 1 + 12.2T + 61T^{2} \)
67 \( 1 - 9.83iT - 67T^{2} \)
71 \( 1 + 8.69T + 71T^{2} \)
73 \( 1 + 15.9T + 73T^{2} \)
79 \( 1 + 14.4iT - 79T^{2} \)
83 \( 1 - 10.1T + 83T^{2} \)
89 \( 1 + 17.1iT - 89T^{2} \)
97 \( 1 - 2.72T + 97T^{2} \)
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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

−10.58548355787065355007991345070, −9.174946077556069487973223369086, −8.793663153321276684619247933123, −7.60978030522156741996447780489, −6.79872989509575151335689952756, −6.00974073030329497129263720302, −4.84517509236701887899121969808, −4.22493869340320264304396149835, −3.14523012911152999281391662581, −1.42519685126304525870748235856, 1.46000512013907201528946323112, 2.79887316072909932478530851121, 3.67239539252027433905649546720, 4.69354530151551858605000783026, 5.84895344239229647221991935251, 6.62626158151990300316072936705, 7.28464915969873556734521588260, 8.857747950496958846516201089755, 9.532708247492045213481293178182, 10.56021564846730938017025554209

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