Properties

Label 2-912-57.50-c1-0-34
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: $\chi_{912} (449, \cdot )$
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

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

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