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Properties

Label 2-927-103.56-c1-0-3
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

\(L(1)\) \(\approx\) \(0.105998 + 1.13530i\)
\(L(\frac12)\) \(\approx\) \(0.105998 + 1.13530i\)
\(L(\frac{3}{2})\) 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

−10.40984504742450652709035992577, −9.736372817485763103448060648400, −8.867103924216797744963870483485, −7.42440563514527036017440648676, −7.10443775092650701405627477918, −6.63695408738934336595514616811, −5.49297442675249214151875756242, −3.99827498328667525324221694539, −3.01401186892533778025348313926, −2.19148184921857957395223147136, 0.52241146284603513314673095874, 1.98663239219093929197143733166, 2.69290671852471596314804609781, 4.59471792440295936085329533189, 5.34251526942899016859699833342, 6.06525110414356579073707727224, 6.89649030486092422416822623732, 8.454961836255232991074236291229, 9.097026793516244803492558488863, 9.546112185832659309511938290673

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