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Properties

Label 2-672-168.11-c1-0-11
Degree $2$
Conductor $672$
Sign $0.982 - 0.188i$
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 + 0.493i)3-s + (1.09 + 1.89i)5-s + (0.451 − 2.60i)7-s + (2.51 − 1.63i)9-s + (1.45 + 0.837i)11-s − 1.56i·13-s + (−2.74 − 2.60i)15-s + (−0.278 − 0.160i)17-s + (−2.87 − 4.97i)19-s + (0.537 + 4.55i)21-s + (3.26 + 5.65i)23-s + (0.114 − 0.198i)25-s + (−3.36 + 3.96i)27-s + 7.04·29-s + (7.76 + 4.48i)31-s + ⋯
L(s)  = 1  + (−0.958 + 0.285i)3-s + (0.488 + 0.845i)5-s + (0.170 − 0.985i)7-s + (0.837 − 0.546i)9-s + (0.437 + 0.252i)11-s − 0.432i·13-s + (−0.709 − 0.671i)15-s + (−0.0676 − 0.0390i)17-s + (−0.659 − 1.14i)19-s + (0.117 + 0.993i)21-s + (0.681 + 1.17i)23-s + (0.0229 − 0.0396i)25-s + (−0.646 + 0.762i)27-s + 1.30·29-s + (1.39 + 0.805i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.188i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 - 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.982 - 0.188i$
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: \((2,\ 672,\ (\ :1/2),\ 0.982 - 0.188i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26555 + 0.120380i\)
\(L(\frac12)\) \(\approx\) \(1.26555 + 0.120380i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.66 - 0.493i)T \)
7 \( 1 + (-0.451 + 2.60i)T \)
good5 \( 1 + (-1.09 - 1.89i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.45 - 0.837i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.56iT - 13T^{2} \)
17 \( 1 + (0.278 + 0.160i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.87 + 4.97i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.26 - 5.65i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.04T + 29T^{2} \)
31 \( 1 + (-7.76 - 4.48i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.946 + 0.546i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 3.44iT - 41T^{2} \)
43 \( 1 - 11.6T + 43T^{2} \)
47 \( 1 + (-2.25 - 3.90i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.42 - 7.67i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.37 + 3.10i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.80 + 2.19i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.716 - 1.24i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.37T + 71T^{2} \)
73 \( 1 + (4.49 - 7.77i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.58 + 2.07i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 9.06iT - 83T^{2} \)
89 \( 1 + (-4.57 + 2.64i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 4.23T + 97T^{2} \)
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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

−10.65067438052112579374656976473, −9.967392569365098419867796232857, −9.047637746933330444654188467343, −7.60607428345784660637800341060, −6.79393330629371334815965874833, −6.25577678369048567540457599444, −4.99794744931052194245462834413, −4.18367968130574883797994907628, −2.85465793280021317871943817650, −1.04101922771862168294472133804, 1.11772078575837535168978741870, 2.37154115504505573980526093247, 4.31743447306163495206882182477, 5.08284785578325137829620472059, 6.06500615978195378971839681642, 6.53730810416618329294284843531, 8.015472152129602021917846587247, 8.745501626997243903087725562614, 9.616210322301302814318405194819, 10.54112400353428642845564658909

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