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Properties

Label 2-756-28.27-c1-0-55
Degree $2$
Conductor $756$
Sign $-0.166 + 0.986i$
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 + (2.60 − 2.68i)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 + (0.696 − 0.717i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.166 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.166 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.166 + 0.986i$
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: \((2,\ 756,\ (\ :1/2),\ -0.166 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67684 - 1.98278i\)
\(L(\frac12)\) \(\approx\) \(1.67684 - 1.98278i\)
\(L(\frac{3}{2})\) 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

−10.62200517445949706560451070002, −9.223527059263730440864341465498, −8.558281880843821421273912263369, −7.45763914295527636914996035246, −6.27312447803995630658765717050, −5.50102046819519795404805261318, −4.50054968158058184595302188546, −3.76287380892787356641659710463, −2.32260261519999999014704010229, −1.11677875699783897195870606314, 2.03405309552929370327474782126, 3.15963010151865111323481499182, 4.43946919908821198795897129726, 5.13487416935954448226864656132, 6.05526867765529172811854047715, 7.34296998448937732025153025105, 7.51777887330768339303276660953, 8.693275320375572273649002738992, 9.653615376779506471575607288176, 10.91127894985588926580469222028

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