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Properties

Label 2-672-21.20-c1-0-10
Degree $2$
Conductor $672$
Sign $0.617 - 0.786i$
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  + (1.66 − 0.468i)3-s − 3.33·5-s + (1.56 + 2.13i)7-s + (2.56 − 1.56i)9-s + 4i·11-s + 5.20i·13-s + (−5.56 + 1.56i)15-s + 1.87·17-s + 0.936i·19-s + (3.60 + 2.83i)21-s − 7.12i·23-s + 6.12·25-s + (3.54 − 3.80i)27-s + 7.12i·29-s + 2.39i·31-s + ⋯
L(s)  = 1  + (0.962 − 0.270i)3-s − 1.49·5-s + (0.590 + 0.807i)7-s + (0.853 − 0.520i)9-s + 1.20i·11-s + 1.44i·13-s + (−1.43 + 0.403i)15-s + 0.454·17-s + 0.214i·19-s + (0.786 + 0.617i)21-s − 1.48i·23-s + 1.22·25-s + (0.681 − 0.731i)27-s + 1.32i·29-s + 0.430i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.617 - 0.786i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (545, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ 0.617 - 0.786i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49603 + 0.727332i\)
\(L(\frac12)\) \(\approx\) \(1.49603 + 0.727332i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.66 + 0.468i)T \)
7 \( 1 + (-1.56 - 2.13i)T \)
good5 \( 1 + 3.33T + 5T^{2} \)
11 \( 1 - 4iT - 11T^{2} \)
13 \( 1 - 5.20iT - 13T^{2} \)
17 \( 1 - 1.87T + 17T^{2} \)
19 \( 1 - 0.936iT - 19T^{2} \)
23 \( 1 + 7.12iT - 23T^{2} \)
29 \( 1 - 7.12iT - 29T^{2} \)
31 \( 1 - 2.39iT - 31T^{2} \)
37 \( 1 - 8.24T + 37T^{2} \)
41 \( 1 - 1.87T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + 6.67T + 47T^{2} \)
53 \( 1 - 0.876iT - 53T^{2} \)
59 \( 1 + 7.08T + 59T^{2} \)
61 \( 1 + 8.13iT - 61T^{2} \)
67 \( 1 + 14.2T + 67T^{2} \)
71 \( 1 + 2.24iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 3.12T + 79T^{2} \)
83 \( 1 + 6.25T + 83T^{2} \)
89 \( 1 - 4.79T + 89T^{2} \)
97 \( 1 - 2.92iT - 97T^{2} \)
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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

−10.70168602965466040564370864273, −9.447152462643848294499246310081, −8.814805240680865760121752963220, −7.994062700960494936882817505828, −7.36896874167761317224944854849, −6.50884596002330358840881597693, −4.68341422010836676278104833604, −4.20471420665522042644389263079, −2.89997621229821709263924039760, −1.70617383462470894143107460107, 0.869451254442353996738550115487, 2.97207669123051630595841140596, 3.70605818764156995028699799670, 4.49421352470243340486019893660, 5.77981787816002549795886291753, 7.48486738171589870696182583218, 7.79552387481936559530965005186, 8.318033845495609968538315897814, 9.452998797957126143092135158092, 10.48420937110732853073377864146

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