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Properties

Label 2-672-168.101-c1-0-20
Degree $2$
Conductor $672$
Sign $0.215 + 0.976i$
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  + (−1.69 + 0.366i)3-s + (2.66 − 1.54i)5-s + (1.46 − 2.20i)7-s + (2.73 − 1.24i)9-s + (0.621 − 1.07i)11-s − 5.98·13-s + (−3.95 + 3.58i)15-s + (0.595 − 1.03i)17-s + (−0.614 − 1.06i)19-s + (−1.66 + 4.26i)21-s + (2.56 − 1.48i)23-s + (2.24 − 3.89i)25-s + (−4.16 + 3.10i)27-s − 3.19·29-s + (−1.33 − 0.773i)31-s + ⋯
L(s)  = 1  + (−0.977 + 0.211i)3-s + (1.19 − 0.688i)5-s + (0.552 − 0.833i)7-s + (0.910 − 0.414i)9-s + (0.187 − 0.324i)11-s − 1.66·13-s + (−1.02 + 0.926i)15-s + (0.144 − 0.250i)17-s + (−0.140 − 0.244i)19-s + (−0.363 + 0.931i)21-s + (0.535 − 0.309i)23-s + (0.449 − 0.778i)25-s + (−0.801 + 0.597i)27-s − 0.594·29-s + (−0.240 − 0.138i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.215 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.215 + 0.976i$
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: \((2,\ 672,\ (\ :1/2),\ 0.215 + 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.980804 - 0.787924i\)
\(L(\frac12)\) \(\approx\) \(0.980804 - 0.787924i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.69 - 0.366i)T \)
7 \( 1 + (-1.46 + 2.20i)T \)
good5 \( 1 + (-2.66 + 1.54i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.621 + 1.07i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.98T + 13T^{2} \)
17 \( 1 + (-0.595 + 1.03i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.614 + 1.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.56 + 1.48i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.19T + 29T^{2} \)
31 \( 1 + (1.33 + 0.773i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.334 - 0.193i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 9.44T + 41T^{2} \)
43 \( 1 + 8.29iT - 43T^{2} \)
47 \( 1 + (3.34 + 5.78i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.25 + 9.09i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.22 - 1.86i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.16 - 5.48i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-10.7 - 6.19i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.21iT - 71T^{2} \)
73 \( 1 + (8.92 + 5.15i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.41 - 11.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.22iT - 83T^{2} \)
89 \( 1 + (6.94 + 12.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 17.1iT - 97T^{2} \)
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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

−10.19358345027508846836907541302, −9.729492849244373199069991091959, −8.810874432303504089061996627399, −7.44466844910592986988730344023, −6.75264602147451579691271156345, −5.49357520556100261666387319278, −5.08465069071396187417554508366, −4.07437402055461543081162159060, −2.14435700828980618316530327424, −0.77174559453303076208821049727, 1.73199878893285971233352245862, 2.63169877577279637095686055800, 4.57830839856072067566476295085, 5.43532957328869436883348132632, 6.08870272279986874361545012046, 7.02489814219944980664750334643, 7.83535810170077802977329751964, 9.373253187210092752887828572453, 9.799011858282618168175399304660, 10.75965288984342763175222546011

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