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Properties

Label 2-912-19.9-c1-0-6
Degree $2$
Conductor $912$
Sign $0.714 + 0.700i$
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  + (−0.766 + 0.642i)3-s + (−3.20 − 1.16i)5-s + (−2.43 + 4.22i)7-s + (0.173 − 0.984i)9-s + (−1.70 − 2.95i)11-s + (2.08 + 1.74i)13-s + (3.20 − 1.16i)15-s + (0.205 + 1.16i)17-s + (2.52 − 3.55i)19-s + (−0.847 − 4.80i)21-s + (−3.20 + 1.16i)23-s + (5.08 + 4.26i)25-s + (0.500 + 0.866i)27-s + (0.655 − 3.71i)29-s + (3.30 − 5.72i)31-s + ⋯
L(s)  = 1  + (−0.442 + 0.371i)3-s + (−1.43 − 0.521i)5-s + (−0.922 + 1.59i)7-s + (0.0578 − 0.328i)9-s + (−0.514 − 0.890i)11-s + (0.577 + 0.484i)13-s + (0.827 − 0.301i)15-s + (0.0498 + 0.282i)17-s + (0.578 − 0.815i)19-s + (−0.184 − 1.04i)21-s + (−0.668 + 0.243i)23-s + (1.01 + 0.853i)25-s + (0.0962 + 0.166i)27-s + (0.121 − 0.690i)29-s + (0.594 − 1.02i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.714 + 0.700i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.714 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.714 + 0.700i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.714 + 0.700i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.556498 - 0.227264i\)
\(L(\frac12)\) \(\approx\) \(0.556498 - 0.227264i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.766 - 0.642i)T \)
19 \( 1 + (-2.52 + 3.55i)T \)
good5 \( 1 + (3.20 + 1.16i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (2.43 - 4.22i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.70 + 2.95i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.08 - 1.74i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-0.205 - 1.16i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (3.20 - 1.16i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-0.655 + 3.71i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-3.30 + 5.72i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 6.75T + 37T^{2} \)
41 \( 1 + (-5.02 + 4.21i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-3.91 - 1.42i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (0.496 - 2.81i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (0.592 - 0.215i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (2.02 + 11.4i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (6.48 - 2.36i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (0.123 - 0.698i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-7.47 - 2.72i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (9.76 - 8.19i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (0.228 - 0.191i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-7.80 + 13.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.18 - 4.35i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (1.23 + 7.01i)T + (-91.1 + 33.1i)T^{2} \)
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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

−9.825816623275829487781296878544, −9.091101966127240765006556976117, −8.421682713447418783528876703385, −7.63591410775811148253478054797, −6.24128830732548445948723058322, −5.75546015747522872478169867305, −4.61221837863720717239391262922, −3.66984554591541632963818929672, −2.68680374396991678373623382566, −0.41462179744505619210083994590, 0.924187222742347068270676713545, 3.02966724378577809872720063017, 3.85686166189794910304913331572, 4.66652901032970216527205831707, 6.13113737053766122552654961931, 6.99507897889312121655301097111, 7.56270819995492596797064821782, 8.062354253239064282864488799717, 9.601753784945414084890580671559, 10.54121678572200673722026334475

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