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Properties

Label 2-672-12.11-c3-0-11
Degree $2$
Conductor $672$
Sign $0.685 - 0.728i$
Analytic cond. $39.6492$
Root an. cond. $6.29676$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.158 − 5.19i)3-s − 4.20i·5-s − 7i·7-s + (−26.9 + 1.64i)9-s − 48.4·11-s − 4.53·13-s + (−21.8 + 0.665i)15-s + 81.1i·17-s + 38.4i·19-s + (−36.3 + 1.10i)21-s + 28.7·23-s + 107.·25-s + (12.8 + 139. i)27-s + 153. i·29-s − 28.0i·31-s + ⋯
L(s)  = 1  + (−0.0304 − 0.999i)3-s − 0.375i·5-s − 0.377i·7-s + (−0.998 + 0.0608i)9-s − 1.32·11-s − 0.0968·13-s + (−0.375 + 0.0114i)15-s + 1.15i·17-s + 0.464i·19-s + (−0.377 + 0.0115i)21-s + 0.260·23-s + 0.858·25-s + (0.0912 + 0.995i)27-s + 0.979i·29-s − 0.162i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 - 0.728i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.685 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.685 - 0.728i$
Analytic conductor: \(39.6492\)
Root analytic conductor: \(6.29676\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (575, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :3/2),\ 0.685 - 0.728i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8343545688\)
\(L(\frac12)\) \(\approx\) \(0.8343545688\)
\(L(\frac{5}{2})\) 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 \( 1 + (0.158 + 5.19i)T \)
7 \( 1 + 7iT \)
good5 \( 1 + 4.20iT - 125T^{2} \)
11 \( 1 + 48.4T + 1.33e3T^{2} \)
13 \( 1 + 4.53T + 2.19e3T^{2} \)
17 \( 1 - 81.1iT - 4.91e3T^{2} \)
19 \( 1 - 38.4iT - 6.85e3T^{2} \)
23 \( 1 - 28.7T + 1.21e4T^{2} \)
29 \( 1 - 153. iT - 2.43e4T^{2} \)
31 \( 1 + 28.0iT - 2.97e4T^{2} \)
37 \( 1 + 62.7T + 5.06e4T^{2} \)
41 \( 1 + 0.500iT - 6.89e4T^{2} \)
43 \( 1 + 431. iT - 7.95e4T^{2} \)
47 \( 1 - 121.T + 1.03e5T^{2} \)
53 \( 1 - 458. iT - 1.48e5T^{2} \)
59 \( 1 + 307.T + 2.05e5T^{2} \)
61 \( 1 + 727.T + 2.26e5T^{2} \)
67 \( 1 + 173. iT - 3.00e5T^{2} \)
71 \( 1 - 706.T + 3.57e5T^{2} \)
73 \( 1 - 676.T + 3.89e5T^{2} \)
79 \( 1 - 494. iT - 4.93e5T^{2} \)
83 \( 1 - 569.T + 5.71e5T^{2} \)
89 \( 1 - 954. iT - 7.04e5T^{2} \)
97 \( 1 + 624.T + 9.12e5T^{2} \)
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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

−10.56383362747216881814539246369, −9.151336306679146857234956054152, −8.294511097942011779740006110120, −7.65951906757331463686882811726, −6.77258633643132756648089125426, −5.75093332989290981339931113231, −4.92176642827722836590904282672, −3.46424963687311623260272021085, −2.25537988721591835854776280393, −1.07098291764818247819378318482, 0.26080855034150296099364054188, 2.52613765610096673678733136117, 3.16891588690112639746720933632, 4.64909802381115566324557303210, 5.21733602463499929105332977039, 6.28295747562408634838252881735, 7.45403063276158031982358488307, 8.364722493084845238351268987704, 9.307233026610715704848956980018, 9.973324532453159630447078384023

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