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Properties

Label 2-756-28.3-c1-0-9
Degree $2$
Conductor $756$
Sign $-0.969 - 0.245i$
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  + (1.29 + 0.577i)2-s + (1.33 + 1.49i)4-s + (−3.03 + 1.75i)5-s + (−0.151 + 2.64i)7-s + (0.859 + 2.69i)8-s + (−4.93 + 0.509i)10-s + (−2.81 − 1.62i)11-s − 2.17i·13-s + (−1.72 + 3.32i)14-s + (−0.446 + 3.97i)16-s + (−4.04 − 2.33i)17-s + (−0.0375 − 0.0650i)19-s + (−6.66 − 2.19i)20-s + (−2.70 − 3.73i)22-s + (−2.40 + 1.38i)23-s + ⋯
L(s)  = 1  + (0.912 + 0.408i)2-s + (0.666 + 0.745i)4-s + (−1.35 + 0.784i)5-s + (−0.0571 + 0.998i)7-s + (0.303 + 0.952i)8-s + (−1.56 + 0.161i)10-s + (−0.850 − 0.490i)11-s − 0.603i·13-s + (−0.459 + 0.888i)14-s + (−0.111 + 0.993i)16-s + (−0.980 − 0.566i)17-s + (−0.00861 − 0.0149i)19-s + (−1.48 − 0.489i)20-s + (−0.575 − 0.795i)22-s + (−0.501 + 0.289i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 - 0.245i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.969 - 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.969 - 0.245i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.969 - 0.245i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.172647 + 1.38483i\)
\(L(\frac12)\) \(\approx\) \(0.172647 + 1.38483i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.29 - 0.577i)T \)
3 \( 1 \)
7 \( 1 + (0.151 - 2.64i)T \)
good5 \( 1 + (3.03 - 1.75i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.81 + 1.62i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.17iT - 13T^{2} \)
17 \( 1 + (4.04 + 2.33i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.0375 + 0.0650i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.40 - 1.38i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 8.09T + 29T^{2} \)
31 \( 1 + (3.66 - 6.34i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.08 - 8.80i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.20iT - 41T^{2} \)
43 \( 1 - 1.49iT - 43T^{2} \)
47 \( 1 + (0.225 + 0.391i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.69 + 2.93i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.23 - 10.7i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-12.7 + 7.33i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.05 - 4.65i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.8iT - 71T^{2} \)
73 \( 1 + (0.852 + 0.492i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.09 - 1.78i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 0.350T + 83T^{2} \)
89 \( 1 + (-3.06 + 1.76i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.05iT - 97T^{2} \)
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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

−11.18689652423198733846064927419, −10.11046445129524785041442326101, −8.472051888296070232965345169971, −8.160183585414454920539908102111, −7.13743463686500752893953534398, −6.37889927902482986626011636892, −5.32152694188115778105764382707, −4.42970470352013400395777322929, −3.16973850227194822284455809803, −2.68647809237697786136493251252, 0.50182159665087640710050150220, 2.20440563210365695748977553348, 3.82002971492677118152489928451, 4.25018451026691459097511395295, 5.03902987312058476915587606841, 6.39975017361594055734725301105, 7.34887215894141379546899996580, 8.004080014467641389401702481276, 9.162876629036310657004944141588, 10.29449099331943281265026074462

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