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Properties

Label 2-756-9.4-c1-0-0
Degree $2$
Conductor $756$
Sign $-0.999 + 0.0167i$
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  + (−1.97 + 3.41i)5-s + (−0.5 − 0.866i)7-s + (0.471 + 0.816i)11-s + (0.5 − 0.866i)13-s − 5.60·17-s + 1.28·19-s + (−2.33 + 4.03i)23-s + (−5.27 − 9.13i)25-s + (−3.83 − 6.63i)29-s + (−3.91 + 6.77i)31-s + 3.94·35-s − 9.82·37-s + (0.471 − 0.816i)41-s + (−4.63 − 8.02i)43-s + (−2.64 − 4.57i)47-s + ⋯
L(s)  = 1  + (−0.881 + 1.52i)5-s + (−0.188 − 0.327i)7-s + (0.142 + 0.246i)11-s + (0.138 − 0.240i)13-s − 1.35·17-s + 0.294·19-s + (−0.485 + 0.841i)23-s + (−1.05 − 1.82i)25-s + (−0.711 − 1.23i)29-s + (−0.703 + 1.21i)31-s + 0.666·35-s − 1.61·37-s + (0.0736 − 0.127i)41-s + (−0.706 − 1.22i)43-s + (−0.385 − 0.667i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0167i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0167i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.999 + 0.0167i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.999 + 0.0167i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00317348 - 0.379088i\)
\(L(\frac12)\) \(\approx\) \(0.00317348 - 0.379088i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (1.97 - 3.41i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.471 - 0.816i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 5.60T + 17T^{2} \)
19 \( 1 - 1.28T + 19T^{2} \)
23 \( 1 + (2.33 - 4.03i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.83 + 6.63i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.91 - 6.77i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 9.82T + 37T^{2} \)
41 \( 1 + (-0.471 + 0.816i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.63 + 8.02i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.64 + 4.57i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 9.22T + 53T^{2} \)
59 \( 1 + (4.77 - 8.26i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.27 - 9.13i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.858 + 1.48i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.54T + 71T^{2} \)
73 \( 1 - 2.71T + 73T^{2} \)
79 \( 1 + (-8.18 - 14.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.198 + 0.343i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 5.50T + 89T^{2} \)
97 \( 1 + (-6.77 - 11.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.67930854079616310555407835135, −10.23292611208196528142411430043, −9.061382130147697120895239913944, −8.039401239855106057637313428981, −7.07888842445284182353478120295, −6.79843941129296479995408034045, −5.52064804373471982670011037144, −4.05241152674397150649007101481, −3.45251340906088493338573930749, −2.20555669123274766998825540203, 0.18485275232889787345606650173, 1.80525521284261654280776356592, 3.52953455567551157675310852909, 4.46456138682150318556744421009, 5.22245464917787869270493972152, 6.35893398966628785791244811860, 7.45261846966760844119076179577, 8.455524698063848824006613096469, 8.865938994444017581630793310744, 9.656757992316085856891741129569

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