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Properties

Label 2-912-57.8-c1-0-15
Degree $2$
Conductor $912$
Sign $-0.211 - 0.977i$
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  + 1.73i·3-s + (3 + 1.73i)5-s + 2·7-s − 2.99·9-s + 1.73i·11-s + (−3 + 1.73i)13-s + (−2.99 + 5.19i)15-s + (6 + 3.46i)17-s + (−0.5 − 4.33i)19-s + 3.46i·21-s + (3.5 + 6.06i)25-s − 5.19i·27-s + (−3 − 5.19i)29-s + 6.92i·31-s − 2.99·33-s + ⋯
L(s)  = 1  + 0.999i·3-s + (1.34 + 0.774i)5-s + 0.755·7-s − 0.999·9-s + 0.522i·11-s + (−0.832 + 0.480i)13-s + (−0.774 + 1.34i)15-s + (1.45 + 0.840i)17-s + (−0.114 − 0.993i)19-s + 0.755i·21-s + (0.700 + 1.21i)25-s − 0.999i·27-s + (−0.557 − 0.964i)29-s + 1.24i·31-s − 0.522·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.211 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.211 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.211 - 0.977i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.211 - 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27414 + 1.57893i\)
\(L(\frac12)\) \(\approx\) \(1.27414 + 1.57893i\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73iT \)
19 \( 1 + (0.5 + 4.33i)T \)
good5 \( 1 + (-3 - 1.73i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 - 2T + 7T^{2} \)
11 \( 1 - 1.73iT - 11T^{2} \)
13 \( 1 + (3 - 1.73i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-6 - 3.46i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.92iT - 31T^{2} \)
37 \( 1 + 6.92iT - 37T^{2} \)
41 \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4 + 6.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3 + 1.73i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5 + 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.5 - 2.59i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.5 - 4.33i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-12 - 6.92i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 5.19iT - 83T^{2} \)
89 \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.5 - 4.33i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.31248576050724156465789401278, −9.622196553332809591683249842154, −8.988892667303114962930770324535, −7.81588038446192400424350943868, −6.84471438265028161356410146149, −5.76972056195236193060375975629, −5.17220262981596176209381566043, −4.13354026037681857881857609659, −2.84529555653443883205991713069, −1.89951593813214637856779014793, 1.02680604417395116400803775047, 1.93633906340176454225883918200, 3.09331530621532167041057148856, 4.93493525578403374552659297951, 5.55772371190918135799808113706, 6.20079492795371499314818591863, 7.53978987569043003328151191223, 8.009091484864696188554396638999, 9.016989903211559510613731951602, 9.740968113429727365914902380019

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