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Properties

Label 2-726-121.23-c1-0-1
Degree $2$
Conductor $726$
Sign $-0.990 + 0.134i$
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  + (−0.415 + 0.909i)2-s − 3-s + (−0.654 − 0.755i)4-s + (0.120 + 0.0353i)5-s + (0.415 − 0.909i)6-s + (−0.477 − 3.32i)7-s + (0.959 − 0.281i)8-s + 9-s + (−0.0822 + 0.0948i)10-s + (−2.50 + 2.17i)11-s + (0.654 + 0.755i)12-s + (0.834 + 0.962i)13-s + (3.22 + 0.945i)14-s + (−0.120 − 0.0353i)15-s + (−0.142 + 0.989i)16-s + (−0.608 − 0.391i)17-s + ⋯
L(s)  = 1  + (−0.293 + 0.643i)2-s − 0.577·3-s + (−0.327 − 0.377i)4-s + (0.0538 + 0.0158i)5-s + (0.169 − 0.371i)6-s + (−0.180 − 1.25i)7-s + (0.339 − 0.0996i)8-s + 0.333·9-s + (−0.0259 + 0.0300i)10-s + (−0.753 + 0.656i)11-s + (0.189 + 0.218i)12-s + (0.231 + 0.266i)13-s + (0.860 + 0.252i)14-s + (−0.0310 − 0.00913i)15-s + (−0.0355 + 0.247i)16-s + (−0.147 − 0.0949i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.134i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(726\)    =    \(2 \cdot 3 \cdot 11^{2}\)
Sign: $-0.990 + 0.134i$
Analytic conductor: \(5.79713\)
Root analytic conductor: \(2.40772\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{726} (265, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 726,\ (\ :1/2),\ -0.990 + 0.134i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0102191 - 0.150835i\)
\(L(\frac12)\) \(\approx\) \(0.0102191 - 0.150835i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.415 - 0.909i)T \)
3 \( 1 + T \)
11 \( 1 + (2.50 - 2.17i)T \)
good5 \( 1 + (-0.120 - 0.0353i)T + (4.20 + 2.70i)T^{2} \)
7 \( 1 + (0.477 + 3.32i)T + (-6.71 + 1.97i)T^{2} \)
13 \( 1 + (-0.834 - 0.962i)T + (-1.85 + 12.8i)T^{2} \)
17 \( 1 + (0.608 + 0.391i)T + (7.06 + 15.4i)T^{2} \)
19 \( 1 + (1.49 - 0.960i)T + (7.89 - 17.2i)T^{2} \)
23 \( 1 + (0.818 - 5.69i)T + (-22.0 - 6.47i)T^{2} \)
29 \( 1 + (4.16 - 2.67i)T + (12.0 - 26.3i)T^{2} \)
31 \( 1 + (5.71 - 6.59i)T + (-4.41 - 30.6i)T^{2} \)
37 \( 1 + (-6.52 + 7.53i)T + (-5.26 - 36.6i)T^{2} \)
41 \( 1 + (1.32 - 2.89i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (9.21 - 2.70i)T + (36.1 - 23.2i)T^{2} \)
47 \( 1 + (-2.49 - 5.47i)T + (-30.7 + 35.5i)T^{2} \)
53 \( 1 + (0.134 + 0.936i)T + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (5.89 + 12.8i)T + (-38.6 + 44.5i)T^{2} \)
61 \( 1 + (5.86 + 12.8i)T + (-39.9 + 46.1i)T^{2} \)
67 \( 1 + (3.04 - 6.66i)T + (-43.8 - 50.6i)T^{2} \)
71 \( 1 + (7.69 - 4.94i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (1.44 - 10.0i)T + (-70.0 - 20.5i)T^{2} \)
79 \( 1 + (6.18 + 1.81i)T + (66.4 + 42.7i)T^{2} \)
83 \( 1 + (0.840 + 5.84i)T + (-79.6 + 23.3i)T^{2} \)
89 \( 1 + (-11.1 - 7.16i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (11.6 - 3.42i)T + (81.6 - 52.4i)T^{2} \)
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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

−10.70593667741478852655011292290, −9.982702403971503373018887584395, −9.278507045692214544316993463711, −7.931501828261082646410320507352, −7.37004032886870147498570538277, −6.58437086270618693431360002994, −5.59028779433432774691963308993, −4.63423872829633772782885668324, −3.66397472426127912498042516285, −1.64279468512926732507121960868, 0.090207066980720133310271330322, 2.02077956112843028742744064921, 3.05392630524343987564133409499, 4.38626874534077860500284312136, 5.59769865556859490188976515994, 6.08540178007450011149545782204, 7.50856773828547121062716706605, 8.455362044455307362216303125823, 9.128778558344138108360527493732, 10.09378630836653139424142912899

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