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Properties

Label 2-912-57.14-c1-0-23
Degree $2$
Conductor $912$
Sign $-0.0256 + 0.999i$
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  + (−1.26 + 1.18i)3-s + (−1.86 + 2.21i)5-s + (−0.562 − 0.973i)7-s + (0.184 − 2.99i)9-s + (2.70 + 1.56i)11-s + (−5.18 + 0.914i)13-s + (−0.283 − 5.01i)15-s + (0.880 + 2.41i)17-s + (−4.13 + 1.37i)19-s + (1.86 + 0.561i)21-s + (−4.31 − 5.14i)23-s + (−0.589 − 3.34i)25-s + (3.31 + 3.99i)27-s + (1.09 + 0.399i)29-s + (3.90 − 2.25i)31-s + ⋯
L(s)  = 1  + (−0.728 + 0.685i)3-s + (−0.832 + 0.992i)5-s + (−0.212 − 0.367i)7-s + (0.0614 − 0.998i)9-s + (0.816 + 0.471i)11-s + (−1.43 + 0.253i)13-s + (−0.0731 − 1.29i)15-s + (0.213 + 0.586i)17-s + (−0.948 + 0.315i)19-s + (0.406 + 0.122i)21-s + (−0.900 − 1.07i)23-s + (−0.117 − 0.668i)25-s + (0.638 + 0.769i)27-s + (0.203 + 0.0740i)29-s + (0.701 − 0.405i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0256 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0256 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.0256 + 0.999i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.0256 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.130440 - 0.133827i\)
\(L(\frac12)\) \(\approx\) \(0.130440 - 0.133827i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.26 - 1.18i)T \)
19 \( 1 + (4.13 - 1.37i)T \)
good5 \( 1 + (1.86 - 2.21i)T + (-0.868 - 4.92i)T^{2} \)
7 \( 1 + (0.562 + 0.973i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.70 - 1.56i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (5.18 - 0.914i)T + (12.2 - 4.44i)T^{2} \)
17 \( 1 + (-0.880 - 2.41i)T + (-13.0 + 10.9i)T^{2} \)
23 \( 1 + (4.31 + 5.14i)T + (-3.99 + 22.6i)T^{2} \)
29 \( 1 + (-1.09 - 0.399i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-3.90 + 2.25i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 12.0iT - 37T^{2} \)
41 \( 1 + (-1.06 + 6.02i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (2.21 + 1.85i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (-0.377 + 1.03i)T + (-36.0 - 30.2i)T^{2} \)
53 \( 1 + (-5.66 + 4.75i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (-6.41 + 2.33i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (5.58 - 4.68i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-2.42 + 6.64i)T + (-51.3 - 43.0i)T^{2} \)
71 \( 1 + (-3.31 - 2.77i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (1.30 - 7.41i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (4.30 + 0.759i)T + (74.2 + 27.0i)T^{2} \)
83 \( 1 + (12.5 - 7.22i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.38 - 13.5i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (1.47 + 4.04i)T + (-74.3 + 62.3i)T^{2} \)
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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

−10.19912272523902958178741347098, −9.271496287388501898727408755343, −8.129562074360623276227045880566, −6.99932432486481896377608696574, −6.68011715876912646363389300095, −5.50539072879264372436945732528, −4.06578220124758152007285170052, −4.02076241513121246113382709239, −2.39984768237248261140180942510, −0.10629079016150189710428551386, 1.23117220341685105775011866950, 2.77644628601474871681800034642, 4.33434149089641879447729453795, 4.99789490700034916778966870633, 6.00338084998836689791440002537, 6.89618993743702198987070232097, 7.82655921327265169193199663081, 8.433344068973969390290450014985, 9.439825920450779782801954904640, 10.30803767992741395741123833670

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