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Properties

Label 2-672-56.19-c1-0-2
Degree $2$
Conductor $672$
Sign $0.109 - 0.993i$
Analytic cond. $5.36594$
Root an. cond. $2.31645$
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  + (−0.866 + 0.5i)3-s + (0.225 − 0.390i)5-s + (0.458 + 2.60i)7-s + (0.499 − 0.866i)9-s + (0.360 + 0.623i)11-s + 3.48·13-s + 0.451i·15-s + (−3.55 + 2.05i)17-s + (−3.97 − 2.29i)19-s + (−1.69 − 2.02i)21-s + (−0.0459 − 0.0265i)23-s + (2.39 + 4.15i)25-s + 0.999i·27-s + 7.85i·29-s + (4.58 + 7.93i)31-s + ⋯
L(s)  = 1  + (−0.499 + 0.288i)3-s + (0.100 − 0.174i)5-s + (0.173 + 0.984i)7-s + (0.166 − 0.288i)9-s + (0.108 + 0.188i)11-s + 0.967·13-s + 0.116i·15-s + (−0.862 + 0.498i)17-s + (−0.912 − 0.526i)19-s + (−0.370 − 0.442i)21-s + (−0.00957 − 0.00552i)23-s + (0.479 + 0.830i)25-s + 0.192i·27-s + 1.45i·29-s + (0.823 + 1.42i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.109 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.109 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.109 - 0.993i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ 0.109 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.879633 + 0.788136i\)
\(L(\frac12)\) \(\approx\) \(0.879633 + 0.788136i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (-0.458 - 2.60i)T \)
good5 \( 1 + (-0.225 + 0.390i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.360 - 0.623i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.48T + 13T^{2} \)
17 \( 1 + (3.55 - 2.05i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.97 + 2.29i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.0459 + 0.0265i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.85iT - 29T^{2} \)
31 \( 1 + (-4.58 - 7.93i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-7.51 - 4.33i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.94iT - 41T^{2} \)
43 \( 1 + 5.17T + 43T^{2} \)
47 \( 1 + (0.460 - 0.796i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.71 - 1.56i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.86 + 2.80i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.54 - 4.41i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.93 - 8.54i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 11.1iT - 71T^{2} \)
73 \( 1 + (3.33 - 1.92i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8.40 - 4.85i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.53iT - 83T^{2} \)
89 \( 1 + (12.6 + 7.28i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 5.14iT - 97T^{2} \)
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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

−10.92665850111062932169516125417, −9.865717450973270452150770441668, −8.730014804312865728600764227020, −8.571944267105706827919932124039, −6.92279032286417215624642155690, −6.22110027206236846686049176378, −5.24273176029936023632258123949, −4.40793711766335248779401404226, −3.05971686846402919913339721156, −1.57592527086506719646609434878, 0.70912390486810168563666138006, 2.28161691750649592135871087292, 3.90654333109442929217769614732, 4.64464961489939567215503018056, 6.15014921619081745545242568485, 6.50343656678660132046074530448, 7.71156959305088791180268923164, 8.376308029201591335822844674214, 9.614682470819059106988215998273, 10.44302527330143525602738432575

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