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Properties

Label 2-726-11.9-c1-0-10
Degree $2$
Conductor $726$
Sign $0.263 + 0.964i$
Analytic cond. $5.79713$
Root an. cond. $2.40772$
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  + (0.309 − 0.951i)2-s + (−0.809 + 0.587i)3-s + (−0.809 − 0.587i)4-s + (0.427 + 1.31i)5-s + (0.309 + 0.951i)6-s + (−1.30 − 0.951i)7-s + (−0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s + 1.38·10-s + 0.999·12-s + (0.381 − 1.17i)13-s + (−1.30 + 0.951i)14-s + (−1.11 − 0.812i)15-s + (0.309 + 0.951i)16-s + (−1.23 − 3.80i)17-s + (−0.809 − 0.587i)18-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (−0.467 + 0.339i)3-s + (−0.404 − 0.293i)4-s + (0.190 + 0.587i)5-s + (0.126 + 0.388i)6-s + (−0.494 − 0.359i)7-s + (−0.286 + 0.207i)8-s + (0.103 − 0.317i)9-s + 0.437·10-s + 0.288·12-s + (0.105 − 0.326i)13-s + (−0.349 + 0.254i)14-s + (−0.288 − 0.209i)15-s + (0.0772 + 0.237i)16-s + (−0.299 − 0.922i)17-s + (−0.190 − 0.138i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(726\)    =    \(2 \cdot 3 \cdot 11^{2}\)
Sign: $0.263 + 0.964i$
Analytic conductor: \(5.79713\)
Root analytic conductor: \(2.40772\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{726} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 726,\ (\ :1/2),\ 0.263 + 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03628 - 0.791217i\)
\(L(\frac12)\) \(\approx\) \(1.03628 - 0.791217i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 + 0.951i)T \)
3 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 \)
good5 \( 1 + (-0.427 - 1.31i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (1.30 + 0.951i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-0.381 + 1.17i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.23 + 3.80i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-6.23 + 4.53i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 7.70T + 23T^{2} \)
29 \( 1 + (-0.118 - 0.0857i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.73 - 5.34i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (7.23 + 5.25i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-6.23 + 4.53i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 9.23T + 43T^{2} \)
47 \( 1 + (3.23 - 2.35i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-1.11 + 3.44i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-4.30 - 3.13i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (3 + 9.23i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 12.9T + 67T^{2} \)
71 \( 1 + (-1.85 - 5.70i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (4.5 + 3.26i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-0.0450 + 0.138i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-0.190 - 0.587i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 11.2T + 89T^{2} \)
97 \( 1 + (2.28 - 7.02i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.54360713140980710269776842724, −9.443875962265263208629811676704, −8.995052126911752660451448029874, −7.30685678852591403143272250245, −6.76317076218152371605112622186, −5.48405465166992806007925420153, −4.78384008501411484827871246397, −3.44945911382816474075721950808, −2.72451004091377007792270629100, −0.77883836019854026957681634952, 1.31354664325704530964861280764, 3.11293345771153644393799417091, 4.40803616721453297911514299456, 5.43817974965911427772412918980, 6.02841265193559537871489754814, 7.00248240957412038773736001815, 7.83313378187839449717477190091, 8.879808856420863244569310614676, 9.458874498905362796990113564688, 10.56661186709613630852368372709

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