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Properties

Label 2-927-927.781-c1-0-29
Degree $2$
Conductor $927$
Sign $-0.668 - 0.743i$
Analytic cond. $7.40213$
Root an. cond. $2.72068$
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  + (1.16 − 0.451i)2-s + (0.811 + 1.52i)3-s + (−0.321 + 0.292i)4-s + (−0.387 + 1.36i)5-s + (1.63 + 1.41i)6-s + (0.345 − 0.521i)7-s + (−1.35 + 2.72i)8-s + (−1.68 + 2.48i)9-s + (0.163 + 1.76i)10-s + (−3.26 − 2.63i)11-s + (−0.708 − 0.253i)12-s + (−1.39 + 2.10i)13-s + (0.167 − 0.764i)14-s + (−2.39 + 0.513i)15-s + (−0.271 + 2.92i)16-s + (2.40 − 1.28i)17-s + ⋯
L(s)  = 1  + (0.824 − 0.319i)2-s + (0.468 + 0.883i)3-s + (−0.160 + 0.146i)4-s + (−0.173 + 0.609i)5-s + (0.669 + 0.578i)6-s + (0.130 − 0.196i)7-s + (−0.480 + 0.964i)8-s + (−0.560 + 0.828i)9-s + (0.0517 + 0.557i)10-s + (−0.985 − 0.793i)11-s + (−0.204 − 0.0731i)12-s + (−0.387 + 0.584i)13-s + (0.0447 − 0.204i)14-s + (−0.619 + 0.132i)15-s + (−0.0678 + 0.732i)16-s + (0.582 − 0.312i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.668 - 0.743i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.668 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $-0.668 - 0.743i$
Analytic conductor: \(7.40213\)
Root analytic conductor: \(2.72068\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (781, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 927,\ (\ :1/2),\ -0.668 - 0.743i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.738553 + 1.65725i\)
\(L(\frac12)\) \(\approx\) \(0.738553 + 1.65725i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.811 - 1.52i)T \)
103 \( 1 + (-6.96 - 7.38i)T \)
good2 \( 1 + (-1.16 + 0.451i)T + (1.47 - 1.34i)T^{2} \)
5 \( 1 + (0.387 - 1.36i)T + (-4.25 - 2.63i)T^{2} \)
7 \( 1 + (-0.345 + 0.521i)T + (-2.72 - 6.44i)T^{2} \)
11 \( 1 + (3.26 + 2.63i)T + (2.35 + 10.7i)T^{2} \)
13 \( 1 + (1.39 - 2.10i)T + (-5.06 - 11.9i)T^{2} \)
17 \( 1 + (-2.40 + 1.28i)T + (9.39 - 14.1i)T^{2} \)
19 \( 1 + (-0.528 - 1.49i)T + (-14.8 + 11.9i)T^{2} \)
23 \( 1 + (-0.119 - 0.770i)T + (-21.9 + 6.97i)T^{2} \)
29 \( 1 + (1.33 - 4.70i)T + (-24.6 - 15.2i)T^{2} \)
31 \( 1 + (4.89 - 2.25i)T + (20.1 - 23.5i)T^{2} \)
37 \( 1 + (0.948 - 1.25i)T + (-10.1 - 35.5i)T^{2} \)
41 \( 1 + (-3.13 - 0.788i)T + (36.1 + 19.3i)T^{2} \)
43 \( 1 + (3.55 + 4.70i)T + (-11.7 + 41.3i)T^{2} \)
47 \( 1 + (1.41 + 2.45i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.88 + 6.87i)T + (-8.12 - 52.3i)T^{2} \)
59 \( 1 + (-3.43 - 5.18i)T + (-22.9 + 54.3i)T^{2} \)
61 \( 1 + (-7.53 + 4.04i)T + (33.6 - 50.8i)T^{2} \)
67 \( 1 + (0.914 - 1.83i)T + (-40.3 - 53.4i)T^{2} \)
71 \( 1 + (2.89 + 0.728i)T + (62.5 + 33.5i)T^{2} \)
73 \( 1 + (-3.02 - 10.6i)T + (-62.0 + 38.4i)T^{2} \)
79 \( 1 + (-10.5 - 10.8i)T + (-2.43 + 78.9i)T^{2} \)
83 \( 1 + (-6.00 - 12.0i)T + (-50.0 + 66.2i)T^{2} \)
89 \( 1 + (-7.20 - 6.57i)T + (8.21 + 88.6i)T^{2} \)
97 \( 1 + (-5.25 + 3.25i)T + (43.2 - 86.8i)T^{2} \)
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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

−10.59716843105348988786905753152, −9.601525894642398795158592567109, −8.712322790939479151637902877809, −7.999396785793773054126784017804, −7.05609298000465346874098288800, −5.49111669504553883786495674922, −5.09685867127868630674186116527, −3.88025504452911994361800436107, −3.28725389870571136779203473541, −2.41493730679967195216545239885, 0.60000491116690434016036708584, 2.21539313022075421948174345670, 3.39906862582922706295179187814, 4.60032522797435104161825273402, 5.36503664100511539981602879867, 6.16312020290045841925426040454, 7.28262224192864527104540972565, 7.87933835750596880448328264286, 8.826870002668097888461554917038, 9.630981214688036121845428486321

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