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Properties

Label 2-672-168.101-c1-0-5
Degree $2$
Conductor $672$
Sign $-0.686 - 0.726i$
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 + 0.283i)3-s + (−2.24 + 1.29i)5-s + (2.53 + 0.751i)7-s + (2.83 − 0.967i)9-s + (1.63 − 2.83i)11-s − 0.912·13-s + (3.47 − 2.85i)15-s + (−2.39 + 4.14i)17-s + (2.66 + 4.61i)19-s + (−4.54 − 0.565i)21-s + (−4.45 + 2.57i)23-s + (0.870 − 1.50i)25-s + (−4.57 + 2.45i)27-s − 1.35·29-s + (−8.18 − 4.72i)31-s + ⋯
L(s)  = 1  + (−0.986 + 0.163i)3-s + (−1.00 + 0.580i)5-s + (0.958 + 0.284i)7-s + (0.946 − 0.322i)9-s + (0.493 − 0.855i)11-s − 0.253·13-s + (0.897 − 0.737i)15-s + (−0.580 + 1.00i)17-s + (0.611 + 1.05i)19-s + (−0.992 − 0.123i)21-s + (−0.929 + 0.536i)23-s + (0.174 − 0.301i)25-s + (−0.881 + 0.473i)27-s − 0.252·29-s + (−1.47 − 0.848i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.686 - 0.726i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.686 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $-0.686 - 0.726i$
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: \((2,\ 672,\ (\ :1/2),\ -0.686 - 0.726i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.235609 + 0.546970i\)
\(L(\frac12)\) \(\approx\) \(0.235609 + 0.546970i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.70 - 0.283i)T \)
7 \( 1 + (-2.53 - 0.751i)T \)
good5 \( 1 + (2.24 - 1.29i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.63 + 2.83i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.912T + 13T^{2} \)
17 \( 1 + (2.39 - 4.14i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.66 - 4.61i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.45 - 2.57i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.35T + 29T^{2} \)
31 \( 1 + (8.18 + 4.72i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.59 - 0.922i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 5.91T + 41T^{2} \)
43 \( 1 - 8.00iT - 43T^{2} \)
47 \( 1 + (-3.29 - 5.70i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.841 - 1.45i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.50 - 0.867i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.72 - 8.18i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.8 + 6.27i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.603iT - 71T^{2} \)
73 \( 1 + (1.29 + 0.746i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.0625 - 0.108i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.246iT - 83T^{2} \)
89 \( 1 + (1.80 + 3.12i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 5.10iT - 97T^{2} \)
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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

−11.03429774873758121344702994045, −10.26968854347457275588620735367, −9.104420050580762099059982588196, −7.987110042448546472380589137506, −7.43115280237610985005319505631, −6.20646060042328595288792358868, −5.53821305949506578688803044768, −4.26444526996292370143502594674, −3.57943829812769853662028456139, −1.60565586176633658892765212335, 0.37378165891783907454213735590, 1.88474783870166670918152756164, 3.97277242424834090173520181151, 4.73169979307271754192809682499, 5.33970105320800067687140142568, 7.00333095104236600413426040686, 7.25177207348539987485883722579, 8.362730711292805658570514177654, 9.315997267951376969975923918685, 10.40363093990737907111862574075

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