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Properties

Label 2-672-168.101-c1-0-7
Degree $2$
Conductor $672$
Sign $0.0359 - 0.999i$
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  + (1.09 − 1.34i)3-s + (−2.46 + 1.42i)5-s + (1.02 + 2.44i)7-s + (−0.614 − 2.93i)9-s + (−2.42 + 4.20i)11-s − 2.75·13-s + (−0.780 + 4.87i)15-s + (−1.75 + 3.03i)17-s + (3.14 + 5.44i)19-s + (4.39 + 1.29i)21-s + (3.15 − 1.82i)23-s + (1.56 − 2.71i)25-s + (−4.61 − 2.38i)27-s − 3.90·29-s + (0.858 + 0.495i)31-s + ⋯
L(s)  = 1  + (0.630 − 0.776i)3-s + (−1.10 + 0.637i)5-s + (0.385 + 0.922i)7-s + (−0.204 − 0.978i)9-s + (−0.731 + 1.26i)11-s − 0.763·13-s + (−0.201 + 1.25i)15-s + (−0.425 + 0.736i)17-s + (0.721 + 1.24i)19-s + (0.959 + 0.282i)21-s + (0.658 − 0.380i)23-s + (0.313 − 0.542i)25-s + (−0.888 − 0.458i)27-s − 0.725·29-s + (0.154 + 0.0889i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0359 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0359 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.0359 - 0.999i$
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: \((2,\ 672,\ (\ :1/2),\ 0.0359 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.773714 + 0.746368i\)
\(L(\frac12)\) \(\approx\) \(0.773714 + 0.746368i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.09 + 1.34i)T \)
7 \( 1 + (-1.02 - 2.44i)T \)
good5 \( 1 + (2.46 - 1.42i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.42 - 4.20i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.75T + 13T^{2} \)
17 \( 1 + (1.75 - 3.03i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.14 - 5.44i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.15 + 1.82i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.90T + 29T^{2} \)
31 \( 1 + (-0.858 - 0.495i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.06 - 0.614i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.10T + 41T^{2} \)
43 \( 1 + 5.11iT - 43T^{2} \)
47 \( 1 + (-5.61 - 9.72i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.00 + 1.73i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.890 + 0.514i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.24 + 2.15i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.02 - 2.90i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.75iT - 71T^{2} \)
73 \( 1 + (-0.291 - 0.168i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.80 - 4.85i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 0.138iT - 83T^{2} \)
89 \( 1 + (0.580 + 1.00i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 11.0iT - 97T^{2} \)
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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

−10.80386786872169992260788423312, −9.758653652576800425370526084670, −8.820978794066221698325885132824, −7.70931054747136455080465438200, −7.63301153422885956212275283771, −6.52258989855134981583801263212, −5.28469423825077715314387601062, −4.03388645739294608681530018464, −2.87593178724257768764993910077, −1.93137760154911281341022410603, 0.51793585293305897067918351017, 2.76743045611704569789025389416, 3.74206236922479953810067955643, 4.69622027603301920790833980339, 5.29301574727112621814448846556, 7.21292420361772028578626442164, 7.73377584571198539920347982084, 8.592627288368071612468724949509, 9.282978864687951572048690293740, 10.33085824509993426619578227509

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