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Properties

Label 2-672-7.5-c2-0-6
Degree $2$
Conductor $672$
Sign $0.0774 - 0.996i$
Analytic cond. $18.3106$
Root an. cond. $4.27909$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 + 0.866i)3-s + (−2.13 − 1.23i)5-s + (−3.79 − 5.88i)7-s + (1.5 − 2.59i)9-s + (−7.39 − 12.8i)11-s + 11.4i·13-s + 4.27·15-s + (−22.0 + 12.7i)17-s + (28.4 + 16.4i)19-s + (10.7 + 5.53i)21-s + (2.87 − 4.97i)23-s + (−9.44 − 16.3i)25-s + 5.19i·27-s + 51.5·29-s + (23.2 − 13.4i)31-s + ⋯
L(s)  = 1  + (−0.5 + 0.288i)3-s + (−0.427 − 0.246i)5-s + (−0.541 − 0.840i)7-s + (0.166 − 0.288i)9-s + (−0.672 − 1.16i)11-s + 0.884i·13-s + 0.285·15-s + (−1.29 + 0.749i)17-s + (1.49 + 0.865i)19-s + (0.513 + 0.263i)21-s + (0.124 − 0.216i)23-s + (−0.377 − 0.654i)25-s + 0.192i·27-s + 1.77·29-s + (0.750 − 0.433i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0774 - 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0774 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.0774 - 0.996i$
Analytic conductor: \(18.3106\)
Root analytic conductor: \(4.27909\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1),\ 0.0774 - 0.996i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6899296891\)
\(L(\frac12)\) \(\approx\) \(0.6899296891\)
\(L(2)\) 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 \( 1 + (1.5 - 0.866i)T \)
7 \( 1 + (3.79 + 5.88i)T \)
good5 \( 1 + (2.13 + 1.23i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (7.39 + 12.8i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 11.4iT - 169T^{2} \)
17 \( 1 + (22.0 - 12.7i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-28.4 - 16.4i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-2.87 + 4.97i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 51.5T + 841T^{2} \)
31 \( 1 + (-23.2 + 13.4i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (30.3 - 52.6i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 65.1iT - 1.68e3T^{2} \)
43 \( 1 + 54.6T + 1.84e3T^{2} \)
47 \( 1 + (21.7 + 12.5i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-15.0 - 26.0i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (8.80 - 5.08i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-72.8 - 42.0i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-58.0 - 100. i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 25.0T + 5.04e3T^{2} \)
73 \( 1 + (1.00 - 0.578i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (36.2 - 62.8i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 132. iT - 6.88e3T^{2} \)
89 \( 1 + (18.0 + 10.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 30.8iT - 9.40e3T^{2} \)
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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

−10.31802251407968941467493675096, −9.997560071540097683860878873405, −8.646426664091594889204493369329, −8.037910388994447799836796775144, −6.75763800102733393513712003475, −6.18477577442851585987414829786, −4.89178887751846311693154713106, −4.06804143754045729498932658968, −3.02708827877209863847816925604, −1.03907579264168858977624662755, 0.31890600983303957756664714608, 2.26566226984480692552301904457, 3.23752119952164981302660351676, 4.87011854136575564353910103064, 5.39521210398933562822468575495, 6.76200305562476557737931435734, 7.22398173076658741689332318335, 8.297136873167237231803717628763, 9.339027946281235023805910030501, 10.10048438774679337152657446936

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