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Properties

Label 2-756-28.27-c1-0-14
Degree $2$
Conductor $756$
Sign $0.531 - 0.847i$
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  + (−1.16 + 0.802i)2-s + (0.711 − 1.86i)4-s − 0.944i·5-s + (−2.59 − 0.517i)7-s + (0.672 + 2.74i)8-s + (0.758 + 1.09i)10-s + 3.44i·11-s − 2.04i·13-s + (3.43 − 1.48i)14-s + (−2.98 − 2.65i)16-s + 4.37i·17-s + 1.09·19-s + (−1.76 − 0.671i)20-s + (−2.76 − 4.01i)22-s + 3.80i·23-s + ⋯
L(s)  = 1  + (−0.823 + 0.567i)2-s + (0.355 − 0.934i)4-s − 0.422i·5-s + (−0.980 − 0.195i)7-s + (0.237 + 0.971i)8-s + (0.239 + 0.347i)10-s + 1.03i·11-s − 0.568i·13-s + (0.918 − 0.395i)14-s + (−0.747 − 0.664i)16-s + 1.06i·17-s + 0.251·19-s + (−0.394 − 0.150i)20-s + (−0.589 − 0.855i)22-s + 0.793i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.531 - 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.531 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.531 - 0.847i$
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: \((2,\ 756,\ (\ :1/2),\ 0.531 - 0.847i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.749899 + 0.414739i\)
\(L(\frac12)\) \(\approx\) \(0.749899 + 0.414739i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.16 - 0.802i)T \)
3 \( 1 \)
7 \( 1 + (2.59 + 0.517i)T \)
good5 \( 1 + 0.944iT - 5T^{2} \)
11 \( 1 - 3.44iT - 11T^{2} \)
13 \( 1 + 2.04iT - 13T^{2} \)
17 \( 1 - 4.37iT - 17T^{2} \)
19 \( 1 - 1.09T + 19T^{2} \)
23 \( 1 - 3.80iT - 23T^{2} \)
29 \( 1 - 4.94T + 29T^{2} \)
31 \( 1 - 4.40T + 31T^{2} \)
37 \( 1 - 6.97T + 37T^{2} \)
41 \( 1 + 7.38iT - 41T^{2} \)
43 \( 1 - 4.99iT - 43T^{2} \)
47 \( 1 - 9.82T + 47T^{2} \)
53 \( 1 - 1.34T + 53T^{2} \)
59 \( 1 + 9.08T + 59T^{2} \)
61 \( 1 + 9.06iT - 61T^{2} \)
67 \( 1 - 11.7iT - 67T^{2} \)
71 \( 1 + 0.593iT - 71T^{2} \)
73 \( 1 - 15.0iT - 73T^{2} \)
79 \( 1 - 9.27iT - 79T^{2} \)
83 \( 1 - 2.26T + 83T^{2} \)
89 \( 1 - 12.5iT - 89T^{2} \)
97 \( 1 + 18.8iT - 97T^{2} \)
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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

−10.16524931127509511286582540146, −9.653519009709672054791452738533, −8.789673508451636562063672452297, −7.914807656221841162007219429453, −7.07598212110266685794992149081, −6.26462049628304376237816031383, −5.34953547611709102451509036355, −4.19533642458194310006402269516, −2.64830880592596562238096330511, −1.08483978635028889162978258233, 0.71460175316567945535958815405, 2.63750259511281311574306371056, 3.18590196942965124565168087441, 4.51078891102147547862838259744, 6.14767966522287745119413881171, 6.76116025122837467005098676123, 7.73242360616694988499568475581, 8.794486588689095039547487436899, 9.307725576799825409496040873006, 10.24116966749295325752061851375

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