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Properties

Label 2-672-56.37-c1-0-13
Degree $2$
Conductor $672$
Sign $-0.683 + 0.729i$
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  + (−0.866 + 0.5i)3-s + (0.586 + 0.338i)5-s + (−2.23 − 1.41i)7-s + (0.499 − 0.866i)9-s + (−1.44 + 0.835i)11-s + 1.28i·13-s − 0.677·15-s + (−3.18 − 5.51i)17-s + (−2.20 − 1.27i)19-s + (2.64 + 0.105i)21-s + (−0.127 + 0.221i)23-s + (−2.27 − 3.93i)25-s + 0.999i·27-s + 6.27i·29-s + (−2.14 − 3.71i)31-s + ⋯
L(s)  = 1  + (−0.499 + 0.288i)3-s + (0.262 + 0.151i)5-s + (−0.845 − 0.534i)7-s + (0.166 − 0.288i)9-s + (−0.436 + 0.251i)11-s + 0.355i·13-s − 0.174·15-s + (−0.771 − 1.33i)17-s + (−0.506 − 0.292i)19-s + (0.576 + 0.0229i)21-s + (−0.0266 + 0.0461i)23-s + (−0.454 − 0.786i)25-s + 0.192i·27-s + 1.16i·29-s + (−0.385 − 0.666i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.683 + 0.729i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.683 + 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $-0.683 + 0.729i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (625, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ -0.683 + 0.729i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.153225 - 0.353374i\)
\(L(\frac12)\) \(\approx\) \(0.153225 - 0.353374i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (2.23 + 1.41i)T \)
good5 \( 1 + (-0.586 - 0.338i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.44 - 0.835i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.28iT - 13T^{2} \)
17 \( 1 + (3.18 + 5.51i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.20 + 1.27i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.127 - 0.221i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.27iT - 29T^{2} \)
31 \( 1 + (2.14 + 3.71i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.62 + 3.24i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.43T + 41T^{2} \)
43 \( 1 + 5.48iT - 43T^{2} \)
47 \( 1 + (-4.73 + 8.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.91 + 2.83i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-8.74 + 5.04i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (13.2 + 7.62i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (13.6 - 7.88i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.48T + 71T^{2} \)
73 \( 1 + (1.43 + 2.48i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.09 - 10.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.63iT - 83T^{2} \)
89 \( 1 + (3.40 - 5.89i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 0.477T + 97T^{2} \)
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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

−10.23853165007216266779657659910, −9.482117316574605114265411972825, −8.653333293333959743138686598461, −7.15694491300184034990017579683, −6.79440402057105700463821947267, −5.63982750852730560165134171266, −4.67424926825955672051359809975, −3.61874453803121164934594656910, −2.29546080441297868210094528100, −0.20434256614677375184270682347, 1.80683055757519796866313383605, 3.13731813522653291017719864512, 4.43521352300087330681519699753, 5.76488499104684433196602640624, 6.10778024135681374110307176828, 7.20903722901977784777894637094, 8.305789191502861213280304441625, 9.049401284470673114212601350842, 10.14114922209285697493640525204, 10.68501162870954770513721515118

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