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Properties

Label 2-912-12.11-c1-0-14
Degree $2$
Conductor $912$
Sign $0.871 + 0.490i$
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  + (−0.881 − 1.49i)3-s + 1.27i·5-s − 0.100i·7-s + (−1.44 + 2.62i)9-s − 3.88·11-s + 5.91·13-s + (1.90 − 1.12i)15-s − 2.12i·17-s − i·19-s + (−0.150 + 0.0887i)21-s + 8.22·23-s + 3.36·25-s + (5.19 − 0.165i)27-s + 7.33i·29-s − 2.91i·31-s + ⋯
L(s)  = 1  + (−0.509 − 0.860i)3-s + 0.572i·5-s − 0.0380i·7-s + (−0.481 + 0.876i)9-s − 1.17·11-s + 1.64·13-s + (0.492 − 0.291i)15-s − 0.514i·17-s − 0.229i·19-s + (−0.0327 + 0.0193i)21-s + 1.71·23-s + 0.672·25-s + (0.999 − 0.0318i)27-s + 1.36i·29-s − 0.523i·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.871 + 0.490i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.871 + 0.490i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25704 - 0.329686i\)
\(L(\frac12)\) \(\approx\) \(1.25704 - 0.329686i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.881 + 1.49i)T \)
19 \( 1 + iT \)
good5 \( 1 - 1.27iT - 5T^{2} \)
7 \( 1 + 0.100iT - 7T^{2} \)
11 \( 1 + 3.88T + 11T^{2} \)
13 \( 1 - 5.91T + 13T^{2} \)
17 \( 1 + 2.12iT - 17T^{2} \)
23 \( 1 - 8.22T + 23T^{2} \)
29 \( 1 - 7.33iT - 29T^{2} \)
31 \( 1 + 2.91iT - 31T^{2} \)
37 \( 1 + 3.36T + 37T^{2} \)
41 \( 1 - 0.0327iT - 41T^{2} \)
43 \( 1 + 7.23iT - 43T^{2} \)
47 \( 1 - 7.65T + 47T^{2} \)
53 \( 1 + 13.7iT - 53T^{2} \)
59 \( 1 - 8.10T + 59T^{2} \)
61 \( 1 + 4.04T + 61T^{2} \)
67 \( 1 - 10.5iT - 67T^{2} \)
71 \( 1 + 3.62T + 71T^{2} \)
73 \( 1 - 9.76T + 73T^{2} \)
79 \( 1 - 0.457iT - 79T^{2} \)
83 \( 1 - 2.00T + 83T^{2} \)
89 \( 1 - 11.0iT - 89T^{2} \)
97 \( 1 - 12.7T + 97T^{2} \)
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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

−10.45735337754269144602241760652, −8.947852311899739610837085255420, −8.329260438998278577242057534380, −7.17076321934574748277149874891, −6.84519700890689705677878382702, −5.67645890173631409730184353070, −5.03904426929130566962797301366, −3.42250970777236146250181768731, −2.43485176890828399784837284156, −0.933771629704021937985443580366, 0.990172669333825651347415596596, 2.91276923918705024633079271664, 3.97727972397294071358620197371, 4.91424894361735794366453252426, 5.67527062529230772752731142406, 6.47436887307201339310000280435, 7.79931292373004069372468466031, 8.736581249242928224711087742358, 9.187340649608165165977518861948, 10.49634305558696486487051905418

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