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Properties

Label 2-672-21.20-c3-0-10
Degree $2$
Conductor $672$
Sign $0.193 - 0.981i$
Analytic cond. $39.6492$
Root an. cond. $6.29676$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.62 − 2.37i)3-s − 19.9·5-s + (−14.5 − 11.4i)7-s + (15.7 − 21.9i)9-s − 24.0i·11-s − 19.9i·13-s + (−92.2 + 47.3i)15-s − 63.2·17-s + 141. i·19-s + (−94.4 − 18.6i)21-s + 13.0i·23-s + 273.·25-s + (20.8 − 138. i)27-s − 144. i·29-s + 155. i·31-s + ⋯
L(s)  = 1  + (0.889 − 0.456i)3-s − 1.78·5-s + (−0.784 − 0.619i)7-s + (0.583 − 0.812i)9-s − 0.658i·11-s − 0.424i·13-s + (−1.58 + 0.814i)15-s − 0.902·17-s + 1.70i·19-s + (−0.981 − 0.193i)21-s + 0.118i·23-s + 2.18·25-s + (0.148 − 0.988i)27-s − 0.928i·29-s + 0.902i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.193 - 0.981i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.193 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.193 - 0.981i$
Analytic conductor: \(39.6492\)
Root analytic conductor: \(6.29676\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (545, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :3/2),\ 0.193 - 0.981i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5480250518\)
\(L(\frac12)\) \(\approx\) \(0.5480250518\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.62 + 2.37i)T \)
7 \( 1 + (14.5 + 11.4i)T \)
good5 \( 1 + 19.9T + 125T^{2} \)
11 \( 1 + 24.0iT - 1.33e3T^{2} \)
13 \( 1 + 19.9iT - 2.19e3T^{2} \)
17 \( 1 + 63.2T + 4.91e3T^{2} \)
19 \( 1 - 141. iT - 6.85e3T^{2} \)
23 \( 1 - 13.0iT - 1.21e4T^{2} \)
29 \( 1 + 144. iT - 2.43e4T^{2} \)
31 \( 1 - 155. iT - 2.97e4T^{2} \)
37 \( 1 + 180.T + 5.06e4T^{2} \)
41 \( 1 - 416.T + 6.89e4T^{2} \)
43 \( 1 - 55.9T + 7.95e4T^{2} \)
47 \( 1 + 37.8T + 1.03e5T^{2} \)
53 \( 1 - 654. iT - 1.48e5T^{2} \)
59 \( 1 + 595.T + 2.05e5T^{2} \)
61 \( 1 - 591. iT - 2.26e5T^{2} \)
67 \( 1 - 863.T + 3.00e5T^{2} \)
71 \( 1 + 371. iT - 3.57e5T^{2} \)
73 \( 1 - 319. iT - 3.89e5T^{2} \)
79 \( 1 + 26.5T + 4.93e5T^{2} \)
83 \( 1 + 329.T + 5.71e5T^{2} \)
89 \( 1 + 869.T + 7.04e5T^{2} \)
97 \( 1 - 1.56e3iT - 9.12e5T^{2} \)
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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

−10.35330389689777649590881711471, −9.216582391296878822802859877555, −8.336279302361124280342781419459, −7.76100583142243603349183811785, −7.07335153341060467431162182887, −6.07357176737029426510106135899, −4.24921736158179538785237447007, −3.70722330550327917130896746462, −2.86621792186943204087307264575, −1.00030865951204542126605695071, 0.16571137706811402471385673894, 2.34878793760474254095884891715, 3.30540566538958318304407893134, 4.22167352115051496938306555401, 4.92189276210644227260264431056, 6.75730862352993810544604747527, 7.30276263480291898369696566076, 8.335638893454939023281384904807, 8.985241687021490557578257228710, 9.648758396549452613154733489155

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