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Properties

Label 2-912-912.83-c1-0-51
Degree $2$
Conductor $912$
Sign $-0.192 - 0.981i$
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  + (−0.366 + 1.36i)2-s + (−1.68 + 0.396i)3-s + (−1.73 − i)4-s + (−0.732 + 2.73i)5-s + (0.0760 − 2.44i)6-s − 2·7-s + (2 − 1.99i)8-s + (2.68 − 1.33i)9-s + (−3.46 − 2i)10-s + (3.15 − 3.15i)11-s + (3.31 + i)12-s + (4.53 − 1.21i)13-s + (0.732 − 2.73i)14-s + (0.152 − 4.89i)15-s + (1.99 + 3.46i)16-s + (3.46 − 2i)17-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)2-s + (−0.973 + 0.228i)3-s + (−0.866 − 0.5i)4-s + (−0.327 + 1.22i)5-s + (0.0310 − 0.999i)6-s − 0.755·7-s + (0.707 − 0.707i)8-s + (0.895 − 0.445i)9-s + (−1.09 − 0.632i)10-s + (0.952 − 0.952i)11-s + (0.957 + 0.288i)12-s + (1.25 − 0.336i)13-s + (0.195 − 0.730i)14-s + (0.0392 − 1.26i)15-s + (0.499 + 0.866i)16-s + (0.840 − 0.485i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.192 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.192 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.192 - 0.981i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.192 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.534222 + 0.648930i\)
\(L(\frac12)\) \(\approx\) \(0.534222 + 0.648930i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.366 - 1.36i)T \)
3 \( 1 + (1.68 - 0.396i)T \)
19 \( 1 + (2.94 - 3.21i)T \)
good5 \( 1 + (0.732 - 2.73i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + 2T + 7T^{2} \)
11 \( 1 + (-3.15 + 3.15i)T - 11iT^{2} \)
13 \( 1 + (-4.53 + 1.21i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-3.46 + 2i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-2.00 - 1.15i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.115 - 0.432i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + 10.3iT - 31T^{2} \)
37 \( 1 + (0.683 - 0.683i)T - 37iT^{2} \)
41 \( 1 + (-1.34 - 2.32i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.06 - 7.69i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-1 + 1.73i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-10.4 + 2.79i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (1.58 + 0.423i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-11.7 + 3.15i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-2.36 - 8.84i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-3.73 + 2.15i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (11.5 - 6.65i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.74 + 3.31i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.15 - 4.15i)T + 83iT^{2} \)
89 \( 1 + (6.63 - 11.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.97 - 8.61i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.20615542647984978175290917647, −9.633909893798014510729350728500, −8.562798776065449466683136537355, −7.57198328008736842147574989419, −6.62333779609582591586568929943, −6.22160666007598141107184059006, −5.55760444458856767301395851100, −3.99191164211312175801941234893, −3.43537399709546927717621788792, −0.881016709144955184525807541072, 0.798960572144152338312145982697, 1.72143266509666764312951585196, 3.62758665467683060121348689080, 4.39874727078375827326338955508, 5.21935645416183916820883332035, 6.40171385511402094757498439143, 7.29534849320504513238758174388, 8.635274063164896887677902653346, 8.978227663147242021681734149728, 10.03741404923829946079764153611

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