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Properties

Label 2-912-76.67-c1-0-10
Degree $2$
Conductor $912$
Sign $-0.277 + 0.960i$
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  + (−0.766 + 0.642i)3-s + (−2.67 − 0.972i)5-s + (2.07 + 1.20i)7-s + (0.173 − 0.984i)9-s + (−0.768 + 0.443i)11-s + (−1.53 + 1.82i)13-s + (2.67 − 0.972i)15-s + (0.112 + 0.638i)17-s + (0.564 − 4.32i)19-s + (−2.36 + 0.416i)21-s + (−1.35 − 3.70i)23-s + (2.36 + 1.98i)25-s + (0.500 + 0.866i)27-s + (−6.77 − 1.19i)29-s + (3.97 − 6.88i)31-s + ⋯
L(s)  = 1  + (−0.442 + 0.371i)3-s + (−1.19 − 0.434i)5-s + (0.785 + 0.453i)7-s + (0.0578 − 0.328i)9-s + (−0.231 + 0.133i)11-s + (−0.424 + 0.506i)13-s + (0.689 − 0.251i)15-s + (0.0273 + 0.154i)17-s + (0.129 − 0.991i)19-s + (−0.515 + 0.0909i)21-s + (−0.281 − 0.773i)23-s + (0.473 + 0.396i)25-s + (0.0962 + 0.166i)27-s + (−1.25 − 0.221i)29-s + (0.714 − 1.23i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.277 + 0.960i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.277 + 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.325328 - 0.432600i\)
\(L(\frac12)\) \(\approx\) \(0.325328 - 0.432600i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.766 - 0.642i)T \)
19 \( 1 + (-0.564 + 4.32i)T \)
good5 \( 1 + (2.67 + 0.972i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (-2.07 - 1.20i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.768 - 0.443i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.53 - 1.82i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (-0.112 - 0.638i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (1.35 + 3.70i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (6.77 + 1.19i)T + (27.2 + 9.91i)T^{2} \)
31 \( 1 + (-3.97 + 6.88i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 5.18iT - 37T^{2} \)
41 \( 1 + (4.03 + 4.81i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (-2.15 + 5.92i)T + (-32.9 - 27.6i)T^{2} \)
47 \( 1 + (-0.153 - 0.0271i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (4.67 + 12.8i)T + (-40.6 + 34.0i)T^{2} \)
59 \( 1 + (0.401 + 2.27i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (7.99 - 2.91i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-1.43 + 8.16i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (2.13 + 0.776i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-8.89 + 7.45i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (9.24 - 7.76i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-8.09 - 4.67i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-8.44 + 10.0i)T + (-15.4 - 87.6i)T^{2} \)
97 \( 1 + (2.73 - 0.481i)T + (91.1 - 33.1i)T^{2} \)
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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

−9.835248237072854400919888562002, −8.933828425837777236938264266338, −8.169059710384693268853006690877, −7.45439077310404854620243453972, −6.39288193328910991905137049739, −5.14801733634593026055885867490, −4.61060898240856055631803084757, −3.70046034874862786826605277933, −2.15894155750792956547369965427, −0.28485419144838483283131291311, 1.40786702531277755465588368658, 3.06340331799681690749256362141, 4.07707811727248817958297225255, 5.04951579783926815538391251683, 6.02521642352664894912493020814, 7.27729749762658428210996177061, 7.65931066584362765987069865211, 8.295365801011811947411561284288, 9.616374343221200759263680265468, 10.67990058761560109934694837056

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