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Properties

Label 2-672-168.5-c1-0-11
Degree $2$
Conductor $672$
Sign $0.966 + 0.256i$
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  + (−1.11 + 1.32i)3-s + (0.337 + 0.195i)5-s + (−1.39 − 2.24i)7-s + (−0.529 − 2.95i)9-s + (0.748 + 1.29i)11-s + 3.28·13-s + (−0.634 + 0.232i)15-s + (−1.68 − 2.91i)17-s + (2.56 − 4.43i)19-s + (4.53 + 0.644i)21-s + (4.72 + 2.72i)23-s + (−2.42 − 4.19i)25-s + (4.51 + 2.57i)27-s − 4.13·29-s + (3.60 − 2.07i)31-s + ⋯
L(s)  = 1  + (−0.641 + 0.766i)3-s + (0.151 + 0.0872i)5-s + (−0.527 − 0.849i)7-s + (−0.176 − 0.984i)9-s + (0.225 + 0.390i)11-s + 0.911·13-s + (−0.163 + 0.0599i)15-s + (−0.407 − 0.706i)17-s + (0.587 − 1.01i)19-s + (0.990 + 0.140i)21-s + (0.985 + 0.569i)23-s + (−0.484 − 0.839i)25-s + (0.868 + 0.496i)27-s − 0.768·29-s + (0.647 − 0.373i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 + 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.966 + 0.256i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ 0.966 + 0.256i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14343 - 0.149280i\)
\(L(\frac12)\) \(\approx\) \(1.14343 - 0.149280i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.11 - 1.32i)T \)
7 \( 1 + (1.39 + 2.24i)T \)
good5 \( 1 + (-0.337 - 0.195i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.748 - 1.29i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.28T + 13T^{2} \)
17 \( 1 + (1.68 + 2.91i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.56 + 4.43i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.72 - 2.72i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.13T + 29T^{2} \)
31 \( 1 + (-3.60 + 2.07i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-7.46 - 4.31i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 11.1T + 41T^{2} \)
43 \( 1 - 4.79iT - 43T^{2} \)
47 \( 1 + (-2.51 + 4.34i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.499 + 0.864i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.36 - 0.785i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.40 + 5.90i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.05 - 1.76i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 14.3iT - 71T^{2} \)
73 \( 1 + (-2.76 + 1.59i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.239 + 0.414i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 17.4iT - 83T^{2} \)
89 \( 1 + (-2.54 + 4.41i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 9.00iT - 97T^{2} \)
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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

−10.54754263855235102452103428872, −9.510124542903399770170959501977, −9.234319721505140041445591891651, −7.72369751350048662557839039247, −6.74552726448147471442422499723, −6.04947400416303218877906522520, −4.84368232129346976399746857508, −4.05718607296009922953587089035, −2.95798955990552239925869858484, −0.800536964207138214866170531866, 1.24374542182965726846331833186, 2.61052292141830878387349975518, 3.98134614250087262566711346080, 5.56647266488771618018473319632, 5.92340020067579220782283814091, 6.84257722433543424254256608417, 7.926848579810194372607515524147, 8.760156123609079454801693688144, 9.608392910602802350418992127649, 10.83700929014609229211346690775

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