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Properties

Label 2-912-19.12-c2-0-15
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: $\chi_{912} (145, \cdot )$
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

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

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