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Properties

Label 2-672-168.125-c1-0-23
Degree $2$
Conductor $672$
Sign $0.242 + 0.970i$
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  + (1.68 − 0.420i)3-s − 3.91i·5-s + 2.64·7-s + (2.64 − 1.41i)9-s − 4.55·13-s + (−1.64 − 6.57i)15-s + 0.979·19-s + (4.44 − 1.11i)21-s + 7.48i·23-s − 10.2·25-s + (3.85 − 3.48i)27-s − 10.3i·35-s + (−7.64 + 1.91i)39-s + (−5.53 − 10.3i)45-s + 7.00·49-s + ⋯
L(s)  = 1  + (0.970 − 0.242i)3-s − 1.74i·5-s + 0.999·7-s + (0.881 − 0.471i)9-s − 1.26·13-s + (−0.424 − 1.69i)15-s + 0.224·19-s + (0.970 − 0.242i)21-s + 1.56i·23-s − 2.05·25-s + (0.740 − 0.671i)27-s − 1.74i·35-s + (−1.22 + 0.306i)39-s + (−0.824 − 1.54i)45-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.242 + 0.970i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ 0.242 + 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73764 - 1.35606i\)
\(L(\frac12)\) \(\approx\) \(1.73764 - 1.35606i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.68 + 0.420i)T \)
7 \( 1 - 2.64T \)
good5 \( 1 + 3.91iT - 5T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 4.55T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 0.979T + 19T^{2} \)
23 \( 1 - 7.48iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 5.29iT - 59T^{2} \)
61 \( 1 - 15.6T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 5.65iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 5.29T + 79T^{2} \)
83 \( 1 - 18.1iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−9.888331505872701041682295387125, −9.374319352203642518330110774395, −8.509006867165859484039113312438, −7.928107798311165645940676587496, −7.17343353390264360373710797556, −5.45412528010638925516366488210, −4.81242895810211097907118851193, −3.84979919738859019654083551106, −2.20222838837364690325138536062, −1.17334719512709836318321605632, 2.17100690012923778954199436275, 2.81776403415260052032449934761, 4.01316831370351382276178475354, 5.04855765382418921588430452577, 6.53185768551640714996107135553, 7.34036259128628844401006887119, 7.914642704423832105932547280529, 8.940863910668026927814286591398, 10.06918589303301743692671609726, 10.45840203041725445400998995148

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