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Properties

Label 2-726-11.3-c1-0-12
Degree $2$
Conductor $726$
Sign $0.751 + 0.659i$
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  + (0.809 − 0.587i)2-s + (0.309 + 0.951i)3-s + (0.309 − 0.951i)4-s + (0.809 + 0.587i)6-s + (0.618 − 1.90i)7-s + (−0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s + 12-s + (3.23 − 2.35i)13-s + (−0.618 − 1.90i)14-s + (−0.809 − 0.587i)16-s + (4.85 + 3.52i)17-s + (−0.309 + 0.951i)18-s + (−1.23 − 3.80i)19-s + 1.99·21-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (0.178 + 0.549i)3-s + (0.154 − 0.475i)4-s + (0.330 + 0.239i)6-s + (0.233 − 0.718i)7-s + (−0.109 − 0.336i)8-s + (−0.269 + 0.195i)9-s + 0.288·12-s + (0.897 − 0.652i)13-s + (−0.165 − 0.508i)14-s + (−0.202 − 0.146i)16-s + (1.17 + 0.855i)17-s + (−0.0728 + 0.224i)18-s + (−0.283 − 0.872i)19-s + 0.436·21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(726\)    =    \(2 \cdot 3 \cdot 11^{2}\)
Sign: $0.751 + 0.659i$
Analytic conductor: \(5.79713\)
Root analytic conductor: \(2.40772\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{726} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 726,\ (\ :1/2),\ 0.751 + 0.659i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.18369 - 0.821746i\)
\(L(\frac12)\) \(\approx\) \(2.18369 - 0.821746i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 + 0.587i)T \)
3 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 \)
good5 \( 1 + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.618 + 1.90i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-3.23 + 2.35i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-4.85 - 3.52i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.23 + 3.80i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 + (-1.85 + 5.70i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (6.47 - 4.70i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (3.09 - 9.51i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.85 - 5.70i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + (1.85 + 5.70i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (6.47 + 4.70i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + (4.85 + 3.52i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.618 + 1.90i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (11.3 - 8.22i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-9.70 - 7.05i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + (11.3 - 8.22i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.58184334676833258597138985665, −9.683027940499611113564186008776, −8.619342537836602738500686805130, −7.80030179525623164655129885743, −6.61256741080538933104838822010, −5.63180111574411498964664071850, −4.65909077301325092250978222538, −3.76847061465799220389807289229, −2.87924100241774967693363930666, −1.17573733301682737218594542033, 1.60410758438713962600183543585, 2.96332169183648328335434514459, 3.98462128924497438275608342564, 5.41854128104526192040423780076, 5.85401371088069416135893977942, 7.11325775577203850701273209416, 7.62961232260779895801794235479, 8.825015127657149131143653729145, 9.228116962998312896361507490959, 10.75224350445294670913772081671

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