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Properties

Label 2-672-56.19-c1-0-9
Degree $2$
Conductor $672$
Sign $0.351 + 0.936i$
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 + (2.08 − 3.61i)5-s + (2.39 − 1.12i)7-s + (0.499 − 0.866i)9-s + (0.855 + 1.48i)11-s − 1.54·13-s + 4.17i·15-s + (2.02 − 1.16i)17-s + (−6.09 − 3.52i)19-s + (−1.51 + 2.16i)21-s + (0.406 + 0.234i)23-s + (−6.21 − 10.7i)25-s + 0.999i·27-s + 3.33i·29-s + (−1.58 − 2.73i)31-s + ⋯
L(s)  = 1  + (−0.499 + 0.288i)3-s + (0.933 − 1.61i)5-s + (0.905 − 0.423i)7-s + (0.166 − 0.288i)9-s + (0.257 + 0.446i)11-s − 0.427·13-s + 1.07i·15-s + (0.490 − 0.282i)17-s + (−1.39 − 0.807i)19-s + (−0.330 + 0.473i)21-s + (0.0846 + 0.0488i)23-s + (−1.24 − 2.15i)25-s + 0.192i·27-s + 0.620i·29-s + (−0.284 − 0.491i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.351 + 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.351 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.351 + 0.936i$
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: \((2,\ 672,\ (\ :1/2),\ 0.351 + 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28148 - 0.888192i\)
\(L(\frac12)\) \(\approx\) \(1.28148 - 0.888192i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.39 + 1.12i)T \)
good5 \( 1 + (-2.08 + 3.61i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.855 - 1.48i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.54T + 13T^{2} \)
17 \( 1 + (-2.02 + 1.16i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.09 + 3.52i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.406 - 0.234i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.33iT - 29T^{2} \)
31 \( 1 + (1.58 + 2.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-7.74 - 4.47i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 5.31iT - 41T^{2} \)
43 \( 1 - 3.42T + 43T^{2} \)
47 \( 1 + (-2.95 + 5.11i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.35 - 0.781i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.26 + 3.04i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.55 + 7.89i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.73 + 6.47i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.49iT - 71T^{2} \)
73 \( 1 + (12.5 - 7.26i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.46 - 0.843i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.72iT - 83T^{2} \)
89 \( 1 + (-1.83 - 1.06i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 1.95iT - 97T^{2} \)
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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

−10.21352793872378372056680193637, −9.495035299303273915261360461853, −8.751441249446227943703547661996, −7.87658907474041497675819789928, −6.64424364998111769249867900111, −5.56723868912194765329612259308, −4.80603256006735268177059220381, −4.28905247744699092683339407856, −2.11874760801327617736902049515, −0.941121684294113059470972069513, 1.80018703046675539691505496409, 2.70321127504807673346157178329, 4.16561369212532162647708700879, 5.70016420968434251255399960346, 6.02445041951728629733655827739, 7.06109848247942972635832014687, 7.86863315008848816314713490379, 9.000005284062551375072698776701, 10.13079062162782331330115967656, 10.68102590266737839233230902650

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