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Properties

Label 2-726-121.89-c1-0-10
Degree $2$
Conductor $726$
Sign $0.806 + 0.591i$
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  + (0.142 − 0.989i)2-s − 3-s + (−0.959 − 0.281i)4-s + (0.750 + 1.64i)5-s + (−0.142 + 0.989i)6-s + (3.43 − 2.21i)7-s + (−0.415 + 0.909i)8-s + 9-s + (1.73 − 0.509i)10-s + (−3.02 + 1.36i)11-s + (0.959 + 0.281i)12-s + (0.637 + 0.187i)13-s + (−1.69 − 3.71i)14-s + (−0.750 − 1.64i)15-s + (0.841 + 0.540i)16-s + (−0.367 + 0.424i)17-s + ⋯
L(s)  = 1  + (0.100 − 0.699i)2-s − 0.577·3-s + (−0.479 − 0.140i)4-s + (0.335 + 0.735i)5-s + (−0.0580 + 0.404i)6-s + (1.29 − 0.835i)7-s + (−0.146 + 0.321i)8-s + 0.333·9-s + (0.548 − 0.160i)10-s + (−0.911 + 0.410i)11-s + (0.276 + 0.0813i)12-s + (0.176 + 0.0519i)13-s + (−0.453 − 0.993i)14-s + (−0.193 − 0.424i)15-s + (0.210 + 0.135i)16-s + (−0.0891 + 0.102i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 + 0.591i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.806 + 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(726\)    =    \(2 \cdot 3 \cdot 11^{2}\)
Sign: $0.806 + 0.591i$
Analytic conductor: \(5.79713\)
Root analytic conductor: \(2.40772\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{726} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 726,\ (\ :1/2),\ 0.806 + 0.591i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44205 - 0.472245i\)
\(L(\frac12)\) \(\approx\) \(1.44205 - 0.472245i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.142 + 0.989i)T \)
3 \( 1 + T \)
11 \( 1 + (3.02 - 1.36i)T \)
good5 \( 1 + (-0.750 - 1.64i)T + (-3.27 + 3.77i)T^{2} \)
7 \( 1 + (-3.43 + 2.21i)T + (2.90 - 6.36i)T^{2} \)
13 \( 1 + (-0.637 - 0.187i)T + (10.9 + 7.02i)T^{2} \)
17 \( 1 + (0.367 - 0.424i)T + (-2.41 - 16.8i)T^{2} \)
19 \( 1 + (-2.60 - 3.01i)T + (-2.70 + 18.8i)T^{2} \)
23 \( 1 + (-6.71 - 4.31i)T + (9.55 + 20.9i)T^{2} \)
29 \( 1 + (0.658 + 0.759i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (-9.15 + 2.68i)T + (26.0 - 16.7i)T^{2} \)
37 \( 1 + (6.21 - 1.82i)T + (31.1 - 20.0i)T^{2} \)
41 \( 1 + (-1.27 + 8.87i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (-1.35 + 2.96i)T + (-28.1 - 32.4i)T^{2} \)
47 \( 1 + (0.263 + 1.83i)T + (-45.0 + 13.2i)T^{2} \)
53 \( 1 + (-6.78 + 4.35i)T + (22.0 - 48.2i)T^{2} \)
59 \( 1 + (-1.63 - 11.3i)T + (-56.6 + 16.6i)T^{2} \)
61 \( 1 + (1.55 + 10.8i)T + (-58.5 + 17.1i)T^{2} \)
67 \( 1 + (1.25 - 8.69i)T + (-64.2 - 18.8i)T^{2} \)
71 \( 1 + (-5.72 - 6.61i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-5.92 - 3.80i)T + (30.3 + 66.4i)T^{2} \)
79 \( 1 + (3.81 + 8.36i)T + (-51.7 + 59.7i)T^{2} \)
83 \( 1 + (3.79 - 2.44i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (5.97 - 6.90i)T + (-12.6 - 88.0i)T^{2} \)
97 \( 1 + (5.22 - 11.4i)T + (-63.5 - 73.3i)T^{2} \)
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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

−10.38650965439900479565244313289, −10.00531475067347099015869644374, −8.611164061732287151452886845633, −7.62661049652947730718745596165, −6.91043425807145869704551191250, −5.54923600496254962555707122441, −4.87277559946810050682294602232, −3.80187537737254065154190336919, −2.44309668327497192606610979544, −1.17813202124220098937965772775, 1.11090182311625691366541960261, 2.79185503163691964607715148011, 4.83611624628924499264514229722, 4.94379508216371952688025461796, 5.83368943171325184923718307895, 6.91455407084923360535533857082, 8.017702222674491305977876071190, 8.619121776267460933345071932025, 9.356285525442102427586231405874, 10.61038877581317485298357904352

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