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Properties

Label 2-756-28.27-c1-0-24
Degree $2$
Conductor $756$
Sign $0.0582 - 0.998i$
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  + (0.304 + 1.38i)2-s + (−1.81 + 0.842i)4-s − 0.556i·5-s + (1.25 − 2.33i)7-s + (−1.71 − 2.24i)8-s + (0.769 − 0.169i)10-s + 0.384i·11-s + 4.88i·13-s + (3.60 + 1.01i)14-s + (2.58 − 3.05i)16-s + 5.55i·17-s + 5.79·19-s + (0.469 + 1.01i)20-s + (−0.530 + 0.117i)22-s + 2.42i·23-s + ⋯
L(s)  = 1  + (0.215 + 0.976i)2-s + (−0.907 + 0.421i)4-s − 0.249i·5-s + (0.473 − 0.880i)7-s + (−0.606 − 0.794i)8-s + (0.243 − 0.0537i)10-s + 0.115i·11-s + 1.35i·13-s + (0.962 + 0.272i)14-s + (0.645 − 0.763i)16-s + 1.34i·17-s + 1.32·19-s + (0.104 + 0.225i)20-s + (−0.113 + 0.0249i)22-s + 0.505i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0582 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0582 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.0582 - 0.998i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.0582 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18186 + 1.11489i\)
\(L(\frac12)\) \(\approx\) \(1.18186 + 1.11489i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.304 - 1.38i)T \)
3 \( 1 \)
7 \( 1 + (-1.25 + 2.33i)T \)
good5 \( 1 + 0.556iT - 5T^{2} \)
11 \( 1 - 0.384iT - 11T^{2} \)
13 \( 1 - 4.88iT - 13T^{2} \)
17 \( 1 - 5.55iT - 17T^{2} \)
19 \( 1 - 5.79T + 19T^{2} \)
23 \( 1 - 2.42iT - 23T^{2} \)
29 \( 1 - 6.72T + 29T^{2} \)
31 \( 1 - 8.00T + 31T^{2} \)
37 \( 1 + 4.16T + 37T^{2} \)
41 \( 1 - 7.47iT - 41T^{2} \)
43 \( 1 + 7.80iT - 43T^{2} \)
47 \( 1 + 11.4T + 47T^{2} \)
53 \( 1 + 3.43T + 53T^{2} \)
59 \( 1 - 1.47T + 59T^{2} \)
61 \( 1 + 3.06iT - 61T^{2} \)
67 \( 1 + 1.91iT - 67T^{2} \)
71 \( 1 - 3.10iT - 71T^{2} \)
73 \( 1 - 4.89iT - 73T^{2} \)
79 \( 1 + 6.35iT - 79T^{2} \)
83 \( 1 - 12.9T + 83T^{2} \)
89 \( 1 + 7.07iT - 89T^{2} \)
97 \( 1 - 9.28iT - 97T^{2} \)
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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

−10.33376803554419647115475740460, −9.584858012739828534256927648751, −8.572521456809840582754404652435, −7.955600288753272736040666521029, −6.95607188065448637606571241866, −6.37003514419365773771406402504, −5.04867898753702063531618214136, −4.42838136930742860951813558146, −3.40259901383151320756250035753, −1.31177175425778753555956295516, 0.963076128337859078633296531453, 2.66859364025411405143388308130, 3.14355899920350232489998642761, 4.84566519198824075891796243387, 5.25807953075501281828247038037, 6.43765045151826468668277935568, 7.83330456240982428915370190143, 8.562136108362894764565040282354, 9.479990072538061643275178331301, 10.23129484765392060555443128303

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