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Properties

Label 2-912-76.31-c1-0-18
Degree $2$
Conductor $912$
Sign $-0.998 - 0.0535i$
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  + (−0.5 − 0.866i)3-s + (−1.66 − 2.87i)5-s − 2.71i·7-s + (−0.499 + 0.866i)9-s − 0.985i·11-s + (4.33 + 2.50i)13-s + (−1.66 + 2.87i)15-s + (−2.51 − 4.35i)17-s + (−0.193 − 4.35i)19-s + (−2.35 + 1.35i)21-s + (−3.68 − 2.12i)23-s + (−3.01 + 5.22i)25-s + 0.999·27-s + (5.83 + 3.36i)29-s − 2.32·31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−0.742 − 1.28i)5-s − 1.02i·7-s + (−0.166 + 0.288i)9-s − 0.297i·11-s + (1.20 + 0.694i)13-s + (−0.428 + 0.742i)15-s + (−0.609 − 1.05i)17-s + (−0.0443 − 0.999i)19-s + (−0.513 + 0.296i)21-s + (−0.769 − 0.444i)23-s + (−0.602 + 1.04i)25-s + 0.192·27-s + (1.08 + 0.625i)29-s − 0.416·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0535i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0535i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.998 - 0.0535i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.998 - 0.0535i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0222237 + 0.829933i\)
\(L(\frac12)\) \(\approx\) \(0.0222237 + 0.829933i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (0.193 + 4.35i)T \)
good5 \( 1 + (1.66 + 2.87i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 2.71iT - 7T^{2} \)
11 \( 1 + 0.985iT - 11T^{2} \)
13 \( 1 + (-4.33 - 2.50i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.51 + 4.35i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (3.68 + 2.12i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.83 - 3.36i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.32T + 31T^{2} \)
37 \( 1 - 8.27iT - 37T^{2} \)
41 \( 1 + (9.96 - 5.75i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (9.48 - 5.47i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.41 + 3.70i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.14 - 2.97i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.66 - 2.87i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.62 + 6.28i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.67 + 11.5i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.19 + 3.79i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.52 + 4.37i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.48 - 9.49i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.70iT - 83T^{2} \)
89 \( 1 + (6.10 + 3.52i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.12 - 4.11i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.566172214770319331507750404423, −8.507272419172692619145330050043, −8.244651982483956137956237513727, −6.98069023018960019816859192390, −6.47658937897315631985728635890, −4.95525206620629896070564049006, −4.50071086320388438595832243783, −3.34130265092125179575627573299, −1.46612673850837671748235052364, −0.42692747764386031043589623778, 2.12294335117106953042883536396, 3.44137324420287571874221107412, 3.95814579391137450980945153486, 5.49367604229185796232086437169, 6.14300151085181036609100733318, 7.00413207127498702954553903303, 8.217914471434106110019548377568, 8.614397706326621326286150013141, 10.02859741467784791917498546020, 10.45728783023539013537578338299

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