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Properties

Label 2-927-103.56-c1-0-37
Degree $2$
Conductor $927$
Sign $-0.982 - 0.185i$
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  + (0.315 − 0.546i)2-s + (0.800 + 1.38i)4-s + (−1.85 − 3.21i)5-s + (−1.44 − 2.50i)7-s + 2.27·8-s − 2.33·10-s + (−0.843 − 1.46i)11-s − 5.58·13-s − 1.82·14-s + (−0.885 + 1.53i)16-s + (1.66 + 2.88i)17-s + (−3.35 + 5.81i)19-s + (2.97 − 5.14i)20-s − 1.06·22-s − 5.65·23-s + ⋯
L(s)  = 1  + (0.223 − 0.386i)2-s + (0.400 + 0.693i)4-s + (−0.829 − 1.43i)5-s + (−0.546 − 0.946i)7-s + 0.803·8-s − 0.739·10-s + (−0.254 − 0.440i)11-s − 1.54·13-s − 0.487·14-s + (−0.221 + 0.383i)16-s + (0.404 + 0.700i)17-s + (−0.770 + 1.33i)19-s + (0.664 − 1.15i)20-s − 0.226·22-s − 1.17·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.185i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.982 - 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $-0.982 - 0.185i$
Analytic conductor: \(7.40213\)
Root analytic conductor: \(2.72068\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (262, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 927,\ (\ :1/2),\ -0.982 - 0.185i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0497841 + 0.533215i\)
\(L(\frac12)\) \(\approx\) \(0.0497841 + 0.533215i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 + (-9.01 - 4.66i)T \)
good2 \( 1 + (-0.315 + 0.546i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (1.85 + 3.21i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.44 + 2.50i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.843 + 1.46i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.58T + 13T^{2} \)
17 \( 1 + (-1.66 - 2.88i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.35 - 5.81i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 5.65T + 23T^{2} \)
29 \( 1 + (-4.32 + 7.48i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.54T + 31T^{2} \)
37 \( 1 + 7.63T + 37T^{2} \)
41 \( 1 + (3.89 - 6.75i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.93 + 3.34i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.07 + 3.58i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.50 + 9.53i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.572 + 0.991i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 11.1T + 61T^{2} \)
67 \( 1 + (3.78 + 6.55i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.77 + 11.7i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 1.78T + 73T^{2} \)
79 \( 1 + 10.0T + 79T^{2} \)
83 \( 1 + (-3.33 + 5.77i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 5.12T + 89T^{2} \)
97 \( 1 + (2.09 - 3.62i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.988058499749682395159369079630, −8.378787653518937223494570151668, −8.124943153774360163450459416374, −7.32599390190227826034281002045, −6.19592516275061396476637835526, −4.83058878609913667583284312052, −4.12643808164072063063843570400, −3.41612662231106797401183675383, −1.87955828122828771110112051997, −0.21373457669315516257764101914, 2.40128525778298724095375515096, 2.89885609390303894035774739148, 4.48196495554814630965662777758, 5.36250390525127953635216951776, 6.46067985258398632206192952501, 7.03424912539634607372967102781, 7.56406300553780746778294625715, 8.852206439633306851145384385244, 10.02365849947666763228556270729, 10.31823438236881057917896403128

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