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Properties

Label 2-912-57.50-c1-0-14
Degree $2$
Conductor $912$
Sign $0.367 + 0.930i$
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  + (−0.809 + 1.53i)3-s + (−3.15 + 1.82i)5-s − 1.32·7-s + (−1.69 − 2.47i)9-s + 5.22i·11-s + (−5.77 − 3.33i)13-s + (−0.237 − 6.31i)15-s + (4.40 − 2.54i)17-s + (3.57 + 2.49i)19-s + (1.07 − 2.03i)21-s + (−2.14 − 1.23i)23-s + (4.14 − 7.18i)25-s + (5.16 − 0.583i)27-s + (0.559 − 0.969i)29-s − 0.304i·31-s + ⋯
L(s)  = 1  + (−0.467 + 0.884i)3-s + (−1.41 + 0.815i)5-s − 0.501·7-s + (−0.563 − 0.826i)9-s + 1.57i·11-s + (−1.60 − 0.925i)13-s + (−0.0611 − 1.62i)15-s + (1.06 − 0.616i)17-s + (0.819 + 0.573i)19-s + (0.234 − 0.443i)21-s + (−0.447 − 0.258i)23-s + (0.829 − 1.43i)25-s + (0.993 − 0.112i)27-s + (0.103 − 0.179i)29-s − 0.0546i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.367 + 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.367 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.367 + 0.930i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.367 + 0.930i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.142131 - 0.0966857i\)
\(L(\frac12)\) \(\approx\) \(0.142131 - 0.0966857i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.809 - 1.53i)T \)
19 \( 1 + (-3.57 - 2.49i)T \)
good5 \( 1 + (3.15 - 1.82i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + 1.32T + 7T^{2} \)
11 \( 1 - 5.22iT - 11T^{2} \)
13 \( 1 + (5.77 + 3.33i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-4.40 + 2.54i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (2.14 + 1.23i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.559 + 0.969i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.304iT - 31T^{2} \)
37 \( 1 - 4.66iT - 37T^{2} \)
41 \( 1 + (-2.16 - 3.74i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.93 + 8.54i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-7.04 - 4.06i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.391 + 0.677i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.58 + 4.47i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.21 + 12.5i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.10 + 1.79i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.10 + 3.65i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.88 + 8.45i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.31 - 2.48i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.31iT - 83T^{2} \)
89 \( 1 + (-0.227 + 0.393i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.45 + 5.45i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.965815717036245304873465235756, −9.557099606430225748454381583689, −7.950775324025337197813535380015, −7.46749613052201976293444360995, −6.64926536994683452939211732334, −5.30918034875795915903588750814, −4.58133067768221848676347440883, −3.55298598509801631720567348082, −2.79915601624644647243540018757, −0.10535751086732480791810543231, 1.05025973781379244956502283670, 2.85239708960614342825455863069, 3.96079517592283282865132360581, 5.09103352284672116729636190633, 5.88661951141392315529825845505, 7.08060148668980406798241461035, 7.63955068910789248448478549799, 8.411183053773171689917146035722, 9.202078413796682151260466253594, 10.37356576492447262528739478824

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