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Properties

Label 2-672-168.11-c1-0-12
Degree $2$
Conductor $672$
Sign $0.844 + 0.536i$
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  + (−1.28 − 1.16i)3-s + (0.646 + 1.11i)5-s + (2.42 − 1.05i)7-s + (0.300 + 2.98i)9-s + (−1.60 − 0.923i)11-s − 2.25i·13-s + (0.470 − 2.18i)15-s + (3.89 + 2.24i)17-s + (2.80 + 4.86i)19-s + (−4.34 − 1.46i)21-s + (0.519 + 0.900i)23-s + (1.66 − 2.88i)25-s + (3.08 − 4.18i)27-s + 1.32·29-s + (−3.69 − 2.13i)31-s + ⋯
L(s)  = 1  + (−0.741 − 0.670i)3-s + (0.289 + 0.500i)5-s + (0.917 − 0.397i)7-s + (0.100 + 0.994i)9-s + (−0.482 − 0.278i)11-s − 0.625i·13-s + (0.121 − 0.565i)15-s + (0.944 + 0.545i)17-s + (0.644 + 1.11i)19-s + (−0.947 − 0.320i)21-s + (0.108 + 0.187i)23-s + (0.332 − 0.576i)25-s + (0.593 − 0.805i)27-s + 0.245·29-s + (−0.664 − 0.383i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 + 0.536i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.844 + 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.844 + 0.536i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ 0.844 + 0.536i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31793 - 0.383290i\)
\(L(\frac12)\) \(\approx\) \(1.31793 - 0.383290i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.28 + 1.16i)T \)
7 \( 1 + (-2.42 + 1.05i)T \)
good5 \( 1 + (-0.646 - 1.11i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.60 + 0.923i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.25iT - 13T^{2} \)
17 \( 1 + (-3.89 - 2.24i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.80 - 4.86i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.519 - 0.900i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.32T + 29T^{2} \)
31 \( 1 + (3.69 + 2.13i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-8.18 + 4.72i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 1.39iT - 41T^{2} \)
43 \( 1 - 6.02T + 43T^{2} \)
47 \( 1 + (5.90 + 10.2i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.02 + 10.4i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-9.57 - 5.52i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.65 - 4.41i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.05 - 5.29i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 14.0T + 71T^{2} \)
73 \( 1 + (4.38 - 7.59i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.37 + 1.36i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 4.74iT - 83T^{2} \)
89 \( 1 + (8.31 - 4.79i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 8.73T + 97T^{2} \)
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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

−10.51078149368863284282514753294, −9.957695630728238414498255657983, −8.290155803114788288115621672600, −7.79454892392339149463047670722, −6.95311653271049700403788701175, −5.75262438621015472717937085453, −5.34405327246512777889504867196, −3.88525400489087191381745105790, −2.38410664254171344852319336090, −1.05403441512341129913597205855, 1.19976453063976053702118555585, 2.89800040660961778513780709289, 4.48726356629234862789377805197, 5.03890411110228472431585155315, 5.79619753633591317894288759569, 7.01345568407585158704480543289, 8.010452579199639065487826891810, 9.229993636006953145556472772436, 9.513941329649045610828706875129, 10.75011847329670963398784719600

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