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Properties

Label 2-672-7.3-c2-0-6
Degree $2$
Conductor $672$
Sign $0.849 - 0.527i$
Analytic cond. $18.3106$
Root an. cond. $4.27909$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 − 0.866i)3-s + (−2.62 + 1.51i)5-s + (−6.96 − 0.665i)7-s + (1.5 + 2.59i)9-s + (0.813 − 1.40i)11-s − 19.0i·13-s + 5.24·15-s + (11.7 + 6.77i)17-s + (−22.8 + 13.2i)19-s + (9.87 + 7.03i)21-s + (−13.2 − 23.0i)23-s + (−7.90 + 13.6i)25-s − 5.19i·27-s + 33.6·29-s + (43.1 + 24.8i)31-s + ⋯
L(s)  = 1  + (−0.5 − 0.288i)3-s + (−0.524 + 0.303i)5-s + (−0.995 − 0.0950i)7-s + (0.166 + 0.288i)9-s + (0.0739 − 0.128i)11-s − 1.46i·13-s + 0.349·15-s + (0.690 + 0.398i)17-s + (−1.20 + 0.695i)19-s + (0.470 + 0.334i)21-s + (−0.577 − 1.00i)23-s + (−0.316 + 0.547i)25-s − 0.192i·27-s + 1.16·29-s + (1.39 + 0.803i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.849 - 0.527i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.849 - 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.849 - 0.527i$
Analytic conductor: \(18.3106\)
Root analytic conductor: \(4.27909\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1),\ 0.849 - 0.527i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9266317883\)
\(L(\frac12)\) \(\approx\) \(0.9266317883\)
\(L(2)\) 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 \( 1 + (1.5 + 0.866i)T \)
7 \( 1 + (6.96 + 0.665i)T \)
good5 \( 1 + (2.62 - 1.51i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-0.813 + 1.40i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 19.0iT - 169T^{2} \)
17 \( 1 + (-11.7 - 6.77i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (22.8 - 13.2i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (13.2 + 23.0i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 33.6T + 841T^{2} \)
31 \( 1 + (-43.1 - 24.8i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-25.0 - 43.3i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 75.1iT - 1.68e3T^{2} \)
43 \( 1 - 28.4T + 1.84e3T^{2} \)
47 \( 1 + (24.8 - 14.3i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-38.4 + 66.5i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (65.1 + 37.5i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-39.6 + 22.9i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-20.7 + 35.9i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 73.8T + 5.04e3T^{2} \)
73 \( 1 + (-49.8 - 28.7i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-49.3 - 85.4i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 33.3iT - 6.88e3T^{2} \)
89 \( 1 + (-9.61 + 5.55i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 89.3iT - 9.40e3T^{2} \)
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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

−10.27350370986119836787179067130, −9.916479232924205565963082110992, −8.263294824032652026334498493813, −7.957389883256499520045393016731, −6.50274191702146660562736817887, −6.24767961589857601010288645848, −4.93193750112586045131916045452, −3.69443229175119821503239551021, −2.75588292060229461424533127118, −0.844325386315702776216381418584, 0.51230163037857500621435059612, 2.40136481462465826400725641306, 3.90650877256793888384057542917, 4.48170939481926816298823343170, 5.83058799316504923686028517874, 6.58319270083220861117290589906, 7.47033273556331554791220407027, 8.679783653180403158645901245395, 9.424588701556208638566901985231, 10.14876451870977267406644255554

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