Properties

Label 2-912-57.41-c1-0-3
Degree $2$
Conductor $912$
Sign $-0.761 - 0.648i$
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  + (−1.57 + 0.726i)3-s + (−1.96 − 0.346i)5-s + (−0.910 − 1.57i)7-s + (1.94 − 2.28i)9-s + (4.10 + 2.37i)11-s + (0.151 − 0.415i)13-s + (3.34 − 0.883i)15-s + (−1.07 + 1.28i)17-s + (3.58 − 2.48i)19-s + (2.57 + 1.81i)21-s + (−5.93 + 1.04i)23-s + (−0.952 − 0.346i)25-s + (−1.39 + 5.00i)27-s + (−4.91 + 4.12i)29-s + (−4.88 + 2.82i)31-s + ⋯
L(s)  = 1  + (−0.907 + 0.419i)3-s + (−0.879 − 0.155i)5-s + (−0.344 − 0.596i)7-s + (0.647 − 0.761i)9-s + (1.23 + 0.715i)11-s + (0.0419 − 0.115i)13-s + (0.863 − 0.228i)15-s + (−0.260 + 0.310i)17-s + (0.821 − 0.569i)19-s + (0.562 + 0.396i)21-s + (−1.23 + 0.218i)23-s + (−0.190 − 0.0693i)25-s + (−0.268 + 0.963i)27-s + (−0.913 + 0.766i)29-s + (−0.877 + 0.506i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.761 - 0.648i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.761 - 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.761 - 0.648i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (497, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.761 - 0.648i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.127445 + 0.346194i\)
\(L(\frac12)\) \(\approx\) \(0.127445 + 0.346194i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.57 - 0.726i)T \)
19 \( 1 + (-3.58 + 2.48i)T \)
good5 \( 1 + (1.96 + 0.346i)T + (4.69 + 1.71i)T^{2} \)
7 \( 1 + (0.910 + 1.57i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-4.10 - 2.37i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.151 + 0.415i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (1.07 - 1.28i)T + (-2.95 - 16.7i)T^{2} \)
23 \( 1 + (5.93 - 1.04i)T + (21.6 - 7.86i)T^{2} \)
29 \( 1 + (4.91 - 4.12i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (4.88 - 2.82i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 5.80iT - 37T^{2} \)
41 \( 1 + (-3.75 + 1.36i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (2.15 - 12.2i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (6.92 + 8.25i)T + (-8.16 + 46.2i)T^{2} \)
53 \( 1 + (-0.424 - 2.40i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (3.87 + 3.24i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-1.80 - 10.2i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (5.27 + 6.28i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (0.897 - 5.08i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (13.5 - 4.94i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (-3.23 - 8.88i)T + (-60.5 + 50.7i)T^{2} \)
83 \( 1 + (-0.523 + 0.302i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.07 - 1.48i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-1.64 + 1.96i)T + (-16.8 - 95.5i)T^{2} \)
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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

−10.38878643320810374464025079802, −9.711172343852998208420546627166, −8.935972227792773549579487396922, −7.67213337601767471658889667036, −6.97656854183211876975329887631, −6.18935062368203110378942279293, −5.02050490571532852105067932227, −4.10831606524511559422981685922, −3.56273688142891468455228232186, −1.37122382989215664909076134197, 0.21233180215990438934002184689, 1.86096407377159421741911552170, 3.51473306585293972295409269495, 4.30758502696404370824515578395, 5.73841863216621305530675623793, 6.11952047689741339272330398042, 7.25108307427155197324547892446, 7.85525215363437323513895568908, 8.977689818596657488895039810221, 9.750997295476765129018566513611

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