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Properties

Label 2-912-12.11-c1-0-28
Degree $2$
Conductor $912$
Sign $-0.832 + 0.554i$
Analytic cond. $7.28235$
Root an. cond. $2.69858$
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  + (−1.72 + 0.110i)3-s − 3.67i·5-s − 4.87i·7-s + (2.97 − 0.383i)9-s + 0.458·11-s + 4.96·13-s + (0.407 + 6.35i)15-s − 3.91i·17-s + i·19-s + (0.540 + 8.42i)21-s − 2.77·23-s − 8.50·25-s + (−5.10 + 0.992i)27-s + 4.21i·29-s − 3.33i·31-s + ⋯
L(s)  = 1  + (−0.997 + 0.0639i)3-s − 1.64i·5-s − 1.84i·7-s + (0.991 − 0.127i)9-s + 0.138·11-s + 1.37·13-s + (0.105 + 1.64i)15-s − 0.949i·17-s + 0.229i·19-s + (0.117 + 1.83i)21-s − 0.577·23-s − 1.70·25-s + (−0.981 + 0.190i)27-s + 0.782i·29-s − 0.599i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.832 + 0.554i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.832 + 0.554i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.314013 - 1.03779i\)
\(L(\frac12)\) \(\approx\) \(0.314013 - 1.03779i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.72 - 0.110i)T \)
19 \( 1 - iT \)
good5 \( 1 + 3.67iT - 5T^{2} \)
7 \( 1 + 4.87iT - 7T^{2} \)
11 \( 1 - 0.458T + 11T^{2} \)
13 \( 1 - 4.96T + 13T^{2} \)
17 \( 1 + 3.91iT - 17T^{2} \)
23 \( 1 + 2.77T + 23T^{2} \)
29 \( 1 - 4.21iT - 29T^{2} \)
31 \( 1 + 3.33iT - 31T^{2} \)
37 \( 1 - 6.65T + 37T^{2} \)
41 \( 1 + 9.60iT - 41T^{2} \)
43 \( 1 - 6.41iT - 43T^{2} \)
47 \( 1 - 11.8T + 47T^{2} \)
53 \( 1 - 3.95iT - 53T^{2} \)
59 \( 1 - 2.01T + 59T^{2} \)
61 \( 1 + 6.63T + 61T^{2} \)
67 \( 1 + 1.01iT - 67T^{2} \)
71 \( 1 + 5.71T + 71T^{2} \)
73 \( 1 - 9.95T + 73T^{2} \)
79 \( 1 - 6.29iT - 79T^{2} \)
83 \( 1 + 11.6T + 83T^{2} \)
89 \( 1 - 14.1iT - 89T^{2} \)
97 \( 1 + 12.6T + 97T^{2} \)
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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

−9.803337555817405838789428158413, −9.043061643859559859611580771367, −7.965744175234316753578517975480, −7.21781946643937700109864268495, −6.17785652807981659056792845445, −5.29328175245775718144088079628, −4.31332053508273791158648895208, −3.91731548868376466674330235980, −1.31094939538385736133842411376, −0.67487272593228897912465159867, 1.88644898047841901889768208650, 2.98421544863185424234151378014, 4.14286245616643999701282612922, 5.67029547251707606984654639862, 6.08338878031452527823519010517, 6.65677648803145179847983498804, 7.83922913565042629605815750849, 8.765337962255891124407189091865, 9.815060900699143130545052028436, 10.60731779153868471499706053353

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