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Properties

Label 2-912-57.29-c1-0-31
Degree $2$
Conductor $912$
Sign $0.490 + 0.871i$
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  + (0.0553 + 1.73i)3-s + (0.882 − 2.42i)5-s + (1.58 − 2.74i)7-s + (−2.99 + 0.191i)9-s + (−2.16 + 1.25i)11-s + (2.71 − 3.24i)13-s + (4.24 + 1.39i)15-s + (−1.32 + 0.233i)17-s + (−3.14 − 3.01i)19-s + (4.83 + 2.59i)21-s + (−1.30 − 3.58i)23-s + (−1.27 − 1.06i)25-s + (−0.497 − 5.17i)27-s + (1.32 − 7.49i)29-s + (−6.89 − 3.97i)31-s + ⋯
L(s)  = 1  + (0.0319 + 0.999i)3-s + (0.394 − 1.08i)5-s + (0.598 − 1.03i)7-s + (−0.997 + 0.0638i)9-s + (−0.653 + 0.377i)11-s + (0.754 − 0.898i)13-s + (1.09 + 0.359i)15-s + (−0.320 + 0.0565i)17-s + (−0.722 − 0.691i)19-s + (1.05 + 0.565i)21-s + (−0.272 − 0.747i)23-s + (−0.254 − 0.213i)25-s + (−0.0956 − 0.995i)27-s + (0.245 − 1.39i)29-s + (−1.23 − 0.714i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.490 + 0.871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.490 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.490 + 0.871i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.490 + 0.871i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28653 - 0.751766i\)
\(L(\frac12)\) \(\approx\) \(1.28653 - 0.751766i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0553 - 1.73i)T \)
19 \( 1 + (3.14 + 3.01i)T \)
good5 \( 1 + (-0.882 + 2.42i)T + (-3.83 - 3.21i)T^{2} \)
7 \( 1 + (-1.58 + 2.74i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.16 - 1.25i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.71 + 3.24i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (1.32 - 0.233i)T + (15.9 - 5.81i)T^{2} \)
23 \( 1 + (1.30 + 3.58i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (-1.32 + 7.49i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (6.89 + 3.97i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.10iT - 37T^{2} \)
41 \( 1 + (4.95 - 4.16i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-11.7 - 4.27i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-6.16 - 1.08i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (-3.46 + 1.26i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (1.54 + 8.75i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (0.133 - 0.0485i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-4.48 - 0.791i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (-8.59 - 3.12i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (1.67 - 1.40i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (6.41 + 7.64i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (-12.3 - 7.11i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-12.7 - 10.6i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-0.538 + 0.0949i)T + (91.1 - 33.1i)T^{2} \)
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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

−9.982506214657916829258988635397, −9.172089089764911077523953233673, −8.332904710454367453527840372507, −7.75712106857823207884946424810, −6.28553880545109697692407793835, −5.31375722726694380530095511311, −4.58723760998333344169086323569, −3.92226806087310723218668187717, −2.39584681255513760501852428876, −0.69768542400036483438792146881, 1.77022027099638373468198418704, 2.47388760661996238921658835490, 3.63851543687247838380164703675, 5.38484696566000372260602930546, 5.95684107848704729305264540120, 6.83877715874667440904749749758, 7.56532305078651468430967881648, 8.682682928757124399134002596794, 9.007383096697430503725716999606, 10.64101916826115410472608630004

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