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Properties

Label 2-672-24.11-c1-0-19
Degree $2$
Conductor $672$
Sign $-0.408 + 0.912i$
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  + (−0.618 − 1.61i)3-s + 1.23·5-s + i·7-s + (−2.23 + 2.00i)9-s − 4.47i·11-s − 3.23i·13-s + (−0.763 − 2.00i)15-s − 2i·17-s − 2.76·19-s + (1.61 − 0.618i)21-s + 8.47·23-s − 3.47·25-s + (4.61 + 2.38i)27-s − 8.94i·31-s + (−7.23 + 2.76i)33-s + ⋯
L(s)  = 1  + (−0.356 − 0.934i)3-s + 0.552·5-s + 0.377i·7-s + (−0.745 + 0.666i)9-s − 1.34i·11-s − 0.897i·13-s + (−0.197 − 0.516i)15-s − 0.485i·17-s − 0.634·19-s + (0.353 − 0.134i)21-s + 1.76·23-s − 0.694·25-s + (0.888 + 0.458i)27-s − 1.60i·31-s + (−1.25 + 0.481i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.408 + 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.408 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $-0.408 + 0.912i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ -0.408 + 0.912i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.663557 - 1.02364i\)
\(L(\frac12)\) \(\approx\) \(0.663557 - 1.02364i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.618 + 1.61i)T \)
7 \( 1 - iT \)
good5 \( 1 - 1.23T + 5T^{2} \)
11 \( 1 + 4.47iT - 11T^{2} \)
13 \( 1 + 3.23iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 + 2.76T + 19T^{2} \)
23 \( 1 - 8.47T + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 8.94iT - 31T^{2} \)
37 \( 1 + 2.47iT - 37T^{2} \)
41 \( 1 + 4.47iT - 41T^{2} \)
43 \( 1 + 8.47T + 43T^{2} \)
47 \( 1 + 6.47T + 47T^{2} \)
53 \( 1 - 1.52T + 53T^{2} \)
59 \( 1 - 3.23iT - 59T^{2} \)
61 \( 1 - 1.70iT - 61T^{2} \)
67 \( 1 + 6.94T + 67T^{2} \)
71 \( 1 - 10.9T + 71T^{2} \)
73 \( 1 - 8.47T + 73T^{2} \)
79 \( 1 - 12.9iT - 79T^{2} \)
83 \( 1 + 8.76iT - 83T^{2} \)
89 \( 1 + 6iT - 89T^{2} \)
97 \( 1 - 12.4T + 97T^{2} \)
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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

−10.40799617155989416378896331714, −9.219889570496754142716640030632, −8.429188714035224018798975440303, −7.62053682340410561458658912939, −6.52124110509473423729468183206, −5.80602703005546441285677486945, −5.11269754387412842185460199688, −3.28417849310778778065760574609, −2.23694824916424707895718832613, −0.68265374875713605552763993352, 1.77102092293634023788360117392, 3.32665850844414997549934612555, 4.52410461185967835691823544576, 5.05797169773987507887363006590, 6.39058306080817616759715728943, 6.99616398017840697550945654650, 8.407088126791287776025299103716, 9.317787480058658693518575629897, 9.916852946695383535665553390522, 10.63434963596612091906985635044

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