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Properties

Label 2-756-84.23-c1-0-32
Degree $2$
Conductor $756$
Sign $0.992 - 0.122i$
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  + (0.253 + 1.39i)2-s + (−1.87 + 0.704i)4-s + (−2.47 − 1.43i)5-s + (2.52 + 0.784i)7-s + (−1.45 − 2.42i)8-s + (1.36 − 3.80i)10-s + (−0.887 − 1.53i)11-s − 4.70·13-s + (−0.451 + 3.71i)14-s + (3.00 − 2.63i)16-s + (6.08 − 3.51i)17-s + (6.93 + 4.00i)19-s + (5.64 + 0.930i)20-s + (1.91 − 1.62i)22-s + (2.46 − 4.26i)23-s + ⋯
L(s)  = 1  + (0.179 + 0.983i)2-s + (−0.935 + 0.352i)4-s + (−1.10 − 0.639i)5-s + (0.955 + 0.296i)7-s + (−0.514 − 0.857i)8-s + (0.430 − 1.20i)10-s + (−0.267 − 0.463i)11-s − 1.30·13-s + (−0.120 + 0.992i)14-s + (0.751 − 0.659i)16-s + (1.47 − 0.852i)17-s + (1.59 + 0.918i)19-s + (1.26 + 0.208i)20-s + (0.407 − 0.346i)22-s + (0.513 − 0.889i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.122i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.992 - 0.122i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.992 - 0.122i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20280 + 0.0738115i\)
\(L(\frac12)\) \(\approx\) \(1.20280 + 0.0738115i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.253 - 1.39i)T \)
3 \( 1 \)
7 \( 1 + (-2.52 - 0.784i)T \)
good5 \( 1 + (2.47 + 1.43i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.887 + 1.53i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.70T + 13T^{2} \)
17 \( 1 + (-6.08 + 3.51i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.93 - 4.00i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.46 + 4.26i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.21iT - 29T^{2} \)
31 \( 1 + (-1.31 + 0.757i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.38 + 2.40i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 4.94iT - 41T^{2} \)
43 \( 1 + 7.39iT - 43T^{2} \)
47 \( 1 + (2.78 - 4.82i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.87 + 1.66i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.717 - 1.24i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.30 + 12.6i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.58 - 1.49i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.4T + 71T^{2} \)
73 \( 1 + (-0.0244 - 0.0423i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.15 - 0.668i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.02T + 83T^{2} \)
89 \( 1 + (-5.61 - 3.24i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 1.07T + 97T^{2} \)
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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

−10.07726711166275451154267310279, −9.328681734108473587034495393643, −8.165133745590868129343294799741, −7.88337803170605123400326996357, −7.20992810456042464349139381257, −5.65280646132874846481679778926, −5.07346314459022667956755774534, −4.25983056543332006993663749606, −3.03408355874178215018433535548, −0.70155855441451511745256891962, 1.26982268868369230908547709210, 2.83763028404517630440143919626, 3.64428340146473075148943905226, 4.78996073568770135414059641798, 5.39198418862073303037880415930, 7.37962078421177503854344750079, 7.52194591144265073340447202816, 8.680419560076133015868395968461, 9.820837326502707471628563468287, 10.40353032751284063542031015300

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