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Properties

Label 2-672-56.19-c1-0-8
Degree $2$
Conductor $672$
Sign $0.946 + 0.323i$
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  + (0.866 − 0.5i)3-s + (1.14 − 1.97i)5-s + (1.95 + 1.78i)7-s + (0.499 − 0.866i)9-s + (2.60 + 4.50i)11-s − 1.44·13-s − 2.28i·15-s + (1.71 − 0.992i)17-s + (4.27 + 2.47i)19-s + (2.58 + 0.570i)21-s + (−6.02 − 3.47i)23-s + (−0.101 − 0.175i)25-s − 0.999i·27-s − 5.21i·29-s + (−1.69 − 2.93i)31-s + ⋯
L(s)  = 1  + (0.499 − 0.288i)3-s + (0.510 − 0.883i)5-s + (0.737 + 0.675i)7-s + (0.166 − 0.288i)9-s + (0.784 + 1.35i)11-s − 0.400·13-s − 0.588i·15-s + (0.416 − 0.240i)17-s + (0.981 + 0.566i)19-s + (0.563 + 0.124i)21-s + (−1.25 − 0.725i)23-s + (−0.0203 − 0.0351i)25-s − 0.192i·27-s − 0.967i·29-s + (−0.304 − 0.527i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 + 0.323i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 + 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.946 + 0.323i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ 0.946 + 0.323i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.11731 - 0.351692i\)
\(L(\frac12)\) \(\approx\) \(2.11731 - 0.351692i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (-1.95 - 1.78i)T \)
good5 \( 1 + (-1.14 + 1.97i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.60 - 4.50i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.44T + 13T^{2} \)
17 \( 1 + (-1.71 + 0.992i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.27 - 2.47i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.02 + 3.47i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.21iT - 29T^{2} \)
31 \( 1 + (1.69 + 2.93i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.53 - 1.46i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.20iT - 41T^{2} \)
43 \( 1 + 12.2T + 43T^{2} \)
47 \( 1 + (-1.25 + 2.17i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.15 + 1.82i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-8.59 + 4.96i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.57 - 6.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.73 + 4.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.6iT - 71T^{2} \)
73 \( 1 + (9.13 - 5.27i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.38 + 2.53i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 0.265iT - 83T^{2} \)
89 \( 1 + (-8.23 - 4.75i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 10.7iT - 97T^{2} \)
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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

−10.02504386444116512876502317696, −9.624465844340452955311405778510, −8.727903660745762838908690948150, −7.961779002521833273057891221445, −7.08243917038610286190094626531, −5.84235794946792361112765103896, −4.98470310955860151131219689170, −4.02319191347958319330821702894, −2.32152874335159625682106682663, −1.48798410338705654616822495419, 1.47416346754462723359082305416, 2.98828331027868697304641348683, 3.77057691500117886807180450069, 5.06580630100738241144024647207, 6.12567613343616870467364510253, 7.09861235975296865624444864142, 7.941823928601026363694903168766, 8.837905943409498559858729250784, 9.804716042696039696698408119488, 10.48212783124150390999715831260

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