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Properties

Label 2-672-168.101-c1-0-0
Degree $2$
Conductor $672$
Sign $-0.997 - 0.0698i$
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  + (−1.70 − 0.298i)3-s + (0.337 − 0.195i)5-s + (−1.39 + 2.24i)7-s + (2.82 + 1.01i)9-s + (0.748 − 1.29i)11-s − 3.28·13-s + (−0.634 + 0.232i)15-s + (1.68 − 2.91i)17-s + (−2.56 − 4.43i)19-s + (3.05 − 3.41i)21-s + (−4.72 + 2.72i)23-s + (−2.42 + 4.19i)25-s + (−4.51 − 2.57i)27-s − 4.13·29-s + (3.60 + 2.07i)31-s + ⋯
L(s)  = 1  + (−0.985 − 0.172i)3-s + (0.151 − 0.0872i)5-s + (−0.527 + 0.849i)7-s + (0.940 + 0.339i)9-s + (0.225 − 0.390i)11-s − 0.911·13-s + (−0.163 + 0.0599i)15-s + (0.407 − 0.706i)17-s + (−0.587 − 1.01i)19-s + (0.665 − 0.746i)21-s + (−0.985 + 0.569i)23-s + (−0.484 + 0.839i)25-s + (−0.868 − 0.496i)27-s − 0.768·29-s + (0.647 + 0.373i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0698i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $-0.997 - 0.0698i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ -0.997 - 0.0698i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00119315 + 0.0341170i\)
\(L(\frac12)\) \(\approx\) \(0.00119315 + 0.0341170i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.70 + 0.298i)T \)
7 \( 1 + (1.39 - 2.24i)T \)
good5 \( 1 + (-0.337 + 0.195i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.748 + 1.29i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.28T + 13T^{2} \)
17 \( 1 + (-1.68 + 2.91i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.56 + 4.43i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.72 - 2.72i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.13T + 29T^{2} \)
31 \( 1 + (-3.60 - 2.07i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (7.46 - 4.31i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 11.1T + 41T^{2} \)
43 \( 1 - 4.79iT - 43T^{2} \)
47 \( 1 + (2.51 + 4.34i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.499 - 0.864i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.36 + 0.785i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.40 + 5.90i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.05 - 1.76i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 14.3iT - 71T^{2} \)
73 \( 1 + (-2.76 - 1.59i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.239 - 0.414i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 17.4iT - 83T^{2} \)
89 \( 1 + (2.54 + 4.41i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 9.00iT - 97T^{2} \)
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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

−11.08196067762510528601858081333, −9.940600192403834213265516891315, −9.473368788448171062400516795503, −8.328497319029695484943314536918, −7.17908875554160153439030815224, −6.44825629181918490381838466211, −5.49035436475966103638046285882, −4.85004412927527343253460156092, −3.32636323901423581943368485301, −1.90421881349356810589723604280, 0.01956164724525761461033332316, 1.84396092693196800281420405420, 3.72334949986290907836694185494, 4.44696356596690288481702209684, 5.66784523592304135311328923729, 6.45831478812509343921358758749, 7.23915336212207440673780971505, 8.221064766500949305119588787323, 9.670027415796073137164113543341, 10.18867045722468601016954722336

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