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Properties

Label 2-726-33.2-c1-0-35
Degree $2$
Conductor $726$
Sign $-0.996 - 0.0860i$
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 + (−0.368 − 1.69i)3-s + (−0.809 − 0.587i)4-s + (2.72 − 0.886i)5-s + (−1.72 − 0.172i)6-s + (−0.407 + 0.560i)7-s + (−0.809 + 0.587i)8-s + (−2.72 + 1.24i)9-s − 2.86i·10-s + (−0.697 + 1.58i)12-s + (−6.72 − 2.18i)13-s + (0.407 + 0.560i)14-s + (−2.50 − 4.29i)15-s + (0.309 + 0.951i)16-s + (−0.419 − 1.29i)17-s + (0.341 + 2.98i)18-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (−0.212 − 0.977i)3-s + (−0.404 − 0.293i)4-s + (1.21 − 0.396i)5-s + (−0.703 − 0.0706i)6-s + (−0.153 + 0.211i)7-s + (−0.286 + 0.207i)8-s + (−0.909 + 0.415i)9-s − 0.907i·10-s + (−0.201 + 0.457i)12-s + (−1.86 − 0.606i)13-s + (0.108 + 0.149i)14-s + (−0.646 − 1.10i)15-s + (0.0772 + 0.237i)16-s + (−0.101 − 0.313i)17-s + (0.0804 + 0.702i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0860i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(726\)    =    \(2 \cdot 3 \cdot 11^{2}\)
Sign: $-0.996 - 0.0860i$
Analytic conductor: \(5.79713\)
Root analytic conductor: \(2.40772\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{726} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 726,\ (\ :1/2),\ -0.996 - 0.0860i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0566202 + 1.31367i\)
\(L(\frac12)\) \(\approx\) \(0.0566202 + 1.31367i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.368 + 1.69i)T \)
11 \( 1 \)
good5 \( 1 + (-2.72 + 0.886i)T + (4.04 - 2.93i)T^{2} \)
7 \( 1 + (0.407 - 0.560i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (6.72 + 2.18i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.419 + 1.29i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (2.39 + 3.29i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + 7.41iT - 23T^{2} \)
29 \( 1 + (-4.48 - 3.25i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.20 + 6.79i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (4.38 + 3.18i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-2.25 + 1.63i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 3.51iT - 43T^{2} \)
47 \( 1 + (2.63 + 3.62i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-9.71 - 3.15i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (1.36 - 1.87i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-3.94 + 1.28i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 - 5.53T + 67T^{2} \)
71 \( 1 + (-8.06 + 2.62i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (4.33 - 5.96i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-12.7 - 4.12i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (1.89 + 5.82i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 1.79iT - 89T^{2} \)
97 \( 1 + (2.19 - 6.76i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.03387754533643372910786859444, −9.229291841168284803477817084200, −8.384203974476067551232574162964, −7.20278811655518103110861174215, −6.30871619791298535772111360694, −5.38897505333481993468215697806, −4.69514915655594251261706966713, −2.61330661214889644053103291079, −2.26163131201643347251363803206, −0.61105050308722688649593628641, 2.26488363442852070948564822914, 3.55526950619620455413389813663, 4.73070919878952042063709187805, 5.43630792684133797800751452039, 6.33160095981579838289862964989, 7.09995642438515582850796758991, 8.335418530266780666966825588077, 9.373100184043295354356855358499, 9.948754805254595887283017142406, 10.39919575588654467582129690192

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