Properties

Label 2-672-7.4-c1-0-10
Degree $2$
Conductor $672$
Sign $0.519 + 0.854i$
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  + (−0.5 + 0.866i)3-s + (−0.227 − 0.393i)5-s + (2.16 − 1.51i)7-s + (−0.499 − 0.866i)9-s + (2.89 − 5.01i)11-s − 5.88·13-s + 0.454·15-s + (−1.45 + 2.51i)17-s + (−2.94 − 5.09i)19-s + (0.227 + 2.63i)21-s + (1.45 + 2.51i)23-s + (2.39 − 4.15i)25-s + 0.999·27-s + 3.54·29-s + (2.16 − 3.75i)31-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−0.101 − 0.176i)5-s + (0.819 − 0.572i)7-s + (−0.166 − 0.288i)9-s + (0.873 − 1.51i)11-s − 1.63·13-s + 0.117·15-s + (−0.352 + 0.611i)17-s + (−0.674 − 1.16i)19-s + (0.0496 + 0.575i)21-s + (0.303 + 0.525i)23-s + (0.479 − 0.830i)25-s + 0.192·27-s + 0.658·29-s + (0.389 − 0.674i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.519 + 0.854i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.519 + 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.519 + 0.854i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ 0.519 + 0.854i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09324 - 0.614750i\)
\(L(\frac12)\) \(\approx\) \(1.09324 - 0.614750i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-2.16 + 1.51i)T \)
good5 \( 1 + (0.227 + 0.393i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.89 + 5.01i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.88T + 13T^{2} \)
17 \( 1 + (1.45 - 2.51i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.94 + 5.09i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.45 - 2.51i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.54T + 29T^{2} \)
31 \( 1 + (-2.16 + 3.75i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.85 + 6.67i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 9.58T + 41T^{2} \)
43 \( 1 - 10.7T + 43T^{2} \)
47 \( 1 + (2.45 + 4.25i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.56 - 11.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.896 + 1.55i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.33 + 4.04i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.94 - 6.82i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.909T + 71T^{2} \)
73 \( 1 + (2.60 - 4.50i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.37 - 2.38i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.97T + 83T^{2} \)
89 \( 1 + (-2.45 - 4.25i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 5.79T + 97T^{2} \)
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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

−10.65817597013046762390908688343, −9.407113025508839834508202072447, −8.760554898698027372657552683452, −7.80036839625036506748421483128, −6.79063641406141932394619269169, −5.78469257705750647931345055435, −4.68282486761253215789688117910, −4.06182769229584032342078039341, −2.58745876350191324964081149448, −0.72098421832338440150283413748, 1.64392056228792586546819303860, 2.63402218813191748297696399293, 4.48327230957096945319915433144, 5.00845291865478558062044036231, 6.34319140392880519752054727097, 7.17417696593417048450389981388, 7.82831071397697000067222411506, 8.957027408546834014156172686250, 9.764329623372093609999935298575, 10.69849512346845623804564211338

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