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Properties

Label 2-912-19.7-c1-0-5
Degree $2$
Conductor $912$
Sign $-0.350 - 0.936i$
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  + (0.5 + 0.866i)3-s + (1.61 + 2.80i)5-s + 0.236·7-s + (−0.499 + 0.866i)9-s − 1.23·11-s + (−1.73 + 3.00i)13-s + (−1.61 + 2.80i)15-s + (2 + 3.46i)17-s + (2 − 3.87i)19-s + (0.118 + 0.204i)21-s + (−1.61 + 2.80i)23-s + (−2.73 + 4.73i)25-s − 0.999·27-s + (0.763 − 1.32i)29-s − 4.70·31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (0.723 + 1.25i)5-s + 0.0892·7-s + (−0.166 + 0.288i)9-s − 0.372·11-s + (−0.481 + 0.833i)13-s + (−0.417 + 0.723i)15-s + (0.485 + 0.840i)17-s + (0.458 − 0.888i)19-s + (0.0257 + 0.0446i)21-s + (−0.337 + 0.584i)23-s + (−0.547 + 0.947i)25-s − 0.192·27-s + (0.141 − 0.245i)29-s − 0.845·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.350 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.350 - 0.936i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.350 - 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01631 + 1.46481i\)
\(L(\frac12)\) \(\approx\) \(1.01631 + 1.46481i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-2 + 3.87i)T \)
good5 \( 1 + (-1.61 - 2.80i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 0.236T + 7T^{2} \)
11 \( 1 + 1.23T + 11T^{2} \)
13 \( 1 + (1.73 - 3.00i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (1.61 - 2.80i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.763 + 1.32i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.70T + 31T^{2} \)
37 \( 1 - 7T + 37T^{2} \)
41 \( 1 + (-1.23 - 2.14i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.11 + 5.40i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1 - 1.73i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.09 + 10.5i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.85 - 3.21i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.5 - 9.52i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.118 + 0.204i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.70 - 11.6i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.73 + 3.00i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.35 - 7.54i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8T + 83T^{2} \)
89 \( 1 + (1.85 - 3.21i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.47 + 9.47i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.13399776935186439355640863415, −9.777789779337289564322620719782, −8.836142704416614207796811355792, −7.72755338302268406553574860076, −6.95313215642196970539009319786, −6.05875306575123657779597952658, −5.13550403045609545257108180870, −3.92887413543731129132071031276, −2.90343944155485468094149055854, −1.99200086209394277531004927186, 0.818424559902287417909515217833, 2.03250585942389258428320908531, 3.23780754328352990311882803979, 4.73492290755602753462477433700, 5.41859457540538921849144409051, 6.24277121651706730038955874782, 7.60126922674867054101081898856, 8.041273905611339544678974883222, 9.069286635562052429131356775889, 9.663639648684703617882564489892

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