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Properties

Label 2-912-912.539-c1-0-150
Degree $2$
Conductor $912$
Sign $0.558 - 0.829i$
Analytic cond. $7.28235$
Root an. cond. $2.69858$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.36 − 0.366i)2-s + (−0.396 − 1.68i)3-s + (1.73 + i)4-s + (−2.73 − 0.732i)5-s + (−0.0760 + 2.44i)6-s − 2·7-s + (−1.99 − 2i)8-s + (−2.68 + 1.33i)9-s + (3.46 + 2i)10-s + (−3.15 − 3.15i)11-s + (1 − 3.31i)12-s + (−1.21 − 4.53i)13-s + (2.73 + 0.732i)14-s + (−0.152 + 4.89i)15-s + (1.99 + 3.46i)16-s + (3.46 − 2i)17-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)2-s + (−0.228 − 0.973i)3-s + (0.866 + 0.5i)4-s + (−1.22 − 0.327i)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 + (0.288 − 0.957i)12-s + (−0.336 − 1.25i)13-s + (0.730 + 0.195i)14-s + (−0.0392 + 1.26i)15-s + (0.499 + 0.866i)16-s + (0.840 − 0.485i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.558 - 0.829i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.558 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.558 - 0.829i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (539, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.558 - 0.829i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 + 0.366i)T \)
3 \( 1 + (0.396 + 1.68i)T \)
19 \( 1 + (3.21 + 2.94i)T \)
good5 \( 1 + (2.73 + 0.732i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + 2T + 7T^{2} \)
11 \( 1 + (3.15 + 3.15i)T + 11iT^{2} \)
13 \( 1 + (1.21 + 4.53i)T + (-11.2 + 6.5i)T^{2} \)
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 \( 1 + (-0.432 + 0.115i)T + (25.1 - 14.5i)T^{2} \)
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 \( 1 + (-7.69 - 2.06i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (1 - 1.73i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.79 - 10.4i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.423 - 1.58i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (3.15 + 11.7i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (8.84 - 2.36i)T + (58.0 - 33.5i)T^{2} \)
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

−9.167280540919366752331530406515, −8.349825193458157670571398808940, −7.79193330210769630407125264582, −7.20117498635365761222372105572, −6.16948929462955796612700540766, −5.15219697378136199064599122177, −3.30661862508816229842651980134, −2.79699740003086168419255528777, −0.849573925949213163461252510628, 0, 2.39032063519244667846873227017, 3.62229847191060557871064586240, 4.52829171408505099746614512901, 5.76480637007576594729161865739, 6.70860101594592559890346000016, 7.56971169262797227987439233683, 8.277743322375849822615284850746, 9.295666287573165120771592684982, 9.964057144975292211555937311689

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