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Properties

Label 2-672-168.125-c1-0-9
Degree $2$
Conductor $672$
Sign $0.721 - 0.692i$
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 + 1.41i)3-s − 1.41i·5-s + (2 + 1.73i)7-s + (−1.00 − 2.82i)9-s + 2.44·11-s − 2·13-s + (2.00 + 1.41i)15-s + 7.34·17-s − 4·19-s + (−4.44 + 1.09i)21-s − 1.41i·23-s + 2.99·25-s + (5.00 + 1.41i)27-s + 4.89·29-s + 6.92i·31-s + ⋯
L(s)  = 1  + (−0.577 + 0.816i)3-s − 0.632i·5-s + (0.755 + 0.654i)7-s + (−0.333 − 0.942i)9-s + 0.738·11-s − 0.554·13-s + (0.516 + 0.365i)15-s + 1.78·17-s − 0.917·19-s + (−0.970 + 0.239i)21-s − 0.294i·23-s + 0.599·25-s + (0.962 + 0.272i)27-s + 0.909·29-s + 1.24i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.721 - 0.692i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.721 - 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.721 - 0.692i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ 0.721 - 0.692i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27907 + 0.514732i\)
\(L(\frac12)\) \(\approx\) \(1.27907 + 0.514732i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 1.41i)T \)
7 \( 1 + (-2 - 1.73i)T \)
good5 \( 1 + 1.41iT - 5T^{2} \)
11 \( 1 - 2.44T + 11T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 - 7.34T + 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + 1.41iT - 23T^{2} \)
29 \( 1 - 4.89T + 29T^{2} \)
31 \( 1 - 6.92iT - 31T^{2} \)
37 \( 1 - 10.3iT - 37T^{2} \)
41 \( 1 + 2.44T + 41T^{2} \)
43 \( 1 + 3.46iT - 43T^{2} \)
47 \( 1 - 4.89T + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 5.65iT - 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 - 3.46iT - 67T^{2} \)
71 \( 1 + 1.41iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 11.3iT - 83T^{2} \)
89 \( 1 + 7.34T + 89T^{2} \)
97 \( 1 + 13.8iT - 97T^{2} \)
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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

−10.46778142094764716850562960111, −9.888015808004937163220705538515, −8.789185010352603637076951660982, −8.392335380553096981516387290526, −6.95945973016092746002333350007, −5.86952518531355899661220805073, −5.05807917507294563227197817988, −4.39789330362685361759053785787, −3.06122324906370784720215346803, −1.24283713305579996058223479907, 1.01309558565644993600620613277, 2.35375973808484512840320134712, 3.84723999178753104384391785607, 5.04285796988912106490503312531, 6.03079170470418280997966974973, 6.96607207109702956158380114271, 7.58994448623308271759356605567, 8.376987049455541450489334538155, 9.749788986809605730815432486479, 10.60087604489532428224372571917

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