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Properties

Label 2-912-19.8-c2-0-26
Degree $2$
Conductor $912$
Sign $0.874 - 0.485i$
Analytic cond. $24.8502$
Root an. cond. $4.98499$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.5 + 0.866i)3-s + (−0.764 + 1.32i)5-s + 1.67·7-s + (1.5 + 2.59i)9-s − 13.3·11-s + (14.2 − 8.21i)13-s + (−2.29 + 1.32i)15-s + (10.8 − 18.8i)17-s + (15.2 + 11.3i)19-s + (2.50 + 1.44i)21-s + (−6.82 − 11.8i)23-s + (11.3 + 19.6i)25-s + 5.19i·27-s + (42.6 − 24.6i)29-s + 42.5i·31-s + ⋯
L(s)  = 1  + (0.5 + 0.288i)3-s + (−0.152 + 0.264i)5-s + 0.238·7-s + (0.166 + 0.288i)9-s − 1.21·11-s + (1.09 − 0.632i)13-s + (−0.152 + 0.0882i)15-s + (0.639 − 1.10i)17-s + (0.803 + 0.595i)19-s + (0.119 + 0.0689i)21-s + (−0.296 − 0.513i)23-s + (0.453 + 0.785i)25-s + 0.192i·27-s + (1.46 − 0.848i)29-s + 1.37i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.874 - 0.485i$
Analytic conductor: \(24.8502\)
Root analytic conductor: \(4.98499\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1),\ 0.874 - 0.485i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.323347036\)
\(L(\frac12)\) \(\approx\) \(2.323347036\)
\(L(2)\) 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 \( 1 + (-1.5 - 0.866i)T \)
19 \( 1 + (-15.2 - 11.3i)T \)
good5 \( 1 + (0.764 - 1.32i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 - 1.67T + 49T^{2} \)
11 \( 1 + 13.3T + 121T^{2} \)
13 \( 1 + (-14.2 + 8.21i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (-10.8 + 18.8i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 + (6.82 + 11.8i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-42.6 + 24.6i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 - 42.5iT - 961T^{2} \)
37 \( 1 - 27.6iT - 1.36e3T^{2} \)
41 \( 1 + (-58.0 - 33.5i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-4.45 + 7.71i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-11.8 - 20.5i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (32.3 - 18.6i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (11.7 + 6.80i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-5.79 - 10.0i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-86.0 + 49.6i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (16.1 + 9.30i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (5.33 - 9.24i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-74.6 - 43.1i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 7.23T + 6.88e3T^{2} \)
89 \( 1 + (-81.0 + 46.8i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (118. + 68.6i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−10.05626773863722651020222738224, −9.139689608653485533544287226423, −8.009104249179262751535602098962, −7.84509522865128338286939571412, −6.56897873891826466409262477408, −5.45825087156344347950443452106, −4.69709654402857159058453353038, −3.33014557193805627689758685512, −2.75906397951420361617634491683, −1.05870693406240181496591700139, 0.925378081206644798170276309720, 2.21733805889728692996327631356, 3.37910655365723759015663699629, 4.38997411967131505076172832092, 5.49231757991244512326810615947, 6.39635197904299543900786516238, 7.51558389086085798777524632463, 8.159155970307105208190545373820, 8.811054995196252874478774423230, 9.780824408806008054392854024918

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