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Properties

Label 2-672-21.20-c1-0-6
Degree $2$
Conductor $672$
Sign $0.142 - 0.989i$
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  + (−1.58 + 0.707i)3-s − 1.41·5-s + (2.23 + 1.41i)7-s + (2.00 − 2.23i)9-s − 3.16i·13-s + (2.23 − 1.00i)15-s + 2.82·17-s + 4.24i·19-s + (−4.53 − 0.654i)21-s + 6i·23-s − 2.99·25-s + (−1.58 + 4.94i)27-s + 4.47i·29-s + 5.65i·31-s + (−3.16 − 2.00i)35-s + ⋯
L(s)  = 1  + (−0.912 + 0.408i)3-s − 0.632·5-s + (0.845 + 0.534i)7-s + (0.666 − 0.745i)9-s − 0.877i·13-s + (0.577 − 0.258i)15-s + 0.685·17-s + 0.973i·19-s + (−0.989 − 0.142i)21-s + 1.25i·23-s − 0.599·25-s + (−0.304 + 0.952i)27-s + 0.830i·29-s + 1.01i·31-s + (−0.534 − 0.338i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.142 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.142 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.142 - 0.989i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (545, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ 0.142 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.723888 + 0.626867i\)
\(L(\frac12)\) \(\approx\) \(0.723888 + 0.626867i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.58 - 0.707i)T \)
7 \( 1 + (-2.23 - 1.41i)T \)
good5 \( 1 + 1.41T + 5T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + 3.16iT - 13T^{2} \)
17 \( 1 - 2.82T + 17T^{2} \)
19 \( 1 - 4.24iT - 19T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 - 4.47iT - 29T^{2} \)
31 \( 1 - 5.65iT - 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 - 8.48T + 41T^{2} \)
43 \( 1 - 8.94T + 43T^{2} \)
47 \( 1 + 6.32T + 47T^{2} \)
53 \( 1 - 13.4iT - 53T^{2} \)
59 \( 1 - 3.16T + 59T^{2} \)
61 \( 1 + 3.16iT - 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 8iT - 71T^{2} \)
73 \( 1 - 12.6iT - 73T^{2} \)
79 \( 1 + 13.4T + 79T^{2} \)
83 \( 1 + 3.16T + 83T^{2} \)
89 \( 1 - 11.3T + 89T^{2} \)
97 \( 1 + 18.9iT - 97T^{2} \)
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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

−10.81940691474556140726269748365, −10.02013032957542389741503038788, −9.032274417406166543433046237974, −7.948882676704866144901957916563, −7.34495955953527063529227679403, −5.86579199879405800026452293482, −5.40771734392345320177182932902, −4.31475816764735676458320920034, −3.28046025678454824584829070319, −1.34033437428094932240817782453, 0.65437565394123676918291992286, 2.16933294829108335598226306306, 4.08803300965341202633061791277, 4.68223269900404205310304944905, 5.83727565922184347617439469362, 6.85389972377714019496147213440, 7.59221196503844242617342222036, 8.308172823227334916720132095260, 9.581101954135554776858480228739, 10.56136108135485758930048537403

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