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Properties

Label 2-672-168.5-c1-0-26
Degree $2$
Conductor $672$
Sign $-0.951 - 0.308i$
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  + (0.528 − 1.64i)3-s + (−2.66 − 1.54i)5-s + (1.46 + 2.20i)7-s + (−2.44 − 1.74i)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 + (4.40 − 1.24i)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  + (0.305 − 0.952i)3-s + (−1.19 − 0.688i)5-s + (0.552 + 0.833i)7-s + (−0.813 − 0.581i)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 + (0.962 − 0.272i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 - 0.308i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.951 - 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $-0.951 - 0.308i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ -0.951 - 0.308i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0659716 + 0.416934i\)
\(L(\frac12)\) \(\approx\) \(0.0659716 + 0.416934i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.528 + 1.64i)T \)
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

−9.880368270654294817245538530435, −8.807655579639215959617332860296, −8.200898388646157377455676845911, −7.64518843607609533348227748180, −6.65773684895027820787933957632, −5.38137854078261570061865863193, −4.56356914099764073908954682549, −3.12326355014860858403700665474, −1.96666185739213434971526498337, −0.20362051454309687427929094214, 2.46074470178400471683254584855, 3.63658778668050434341807340094, 4.39651464096111180143584041633, 5.18440858605752670607534818375, 6.86046849199039286463101424513, 7.61921350825046895201254563504, 8.190208350758525101933041829340, 9.404945441128017826718089593282, 10.30088624180586208610648558115, 10.77569059435213083775141415927

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