Properties

Label 2-672-12.11-c1-0-13
Degree $2$
Conductor $672$
Sign $0.169 + 0.985i$
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  + (−1.41 − i)3-s − 1.41i·5-s + i·7-s + (1.00 + 2.82i)9-s + 1.41·11-s + 6·13-s + (−1.41 + 2.00i)15-s − 1.41i·17-s − 4i·19-s + (1 − 1.41i)21-s − 7.07·23-s + 2.99·25-s + (1.41 − 5.00i)27-s − 8.48i·29-s − 2i·31-s + ⋯
L(s)  = 1  + (−0.816 − 0.577i)3-s − 0.632i·5-s + 0.377i·7-s + (0.333 + 0.942i)9-s + 0.426·11-s + 1.66·13-s + (−0.365 + 0.516i)15-s − 0.342i·17-s − 0.917i·19-s + (0.218 − 0.308i)21-s − 1.47·23-s + 0.599·25-s + (0.272 − 0.962i)27-s − 1.57i·29-s − 0.359i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.169 + 0.985i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (575, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ 0.169 + 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.877675 - 0.739914i\)
\(L(\frac12)\) \(\approx\) \(0.877675 - 0.739914i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.41 + i)T \)
7 \( 1 - iT \)
good5 \( 1 + 1.41iT - 5T^{2} \)
11 \( 1 - 1.41T + 11T^{2} \)
13 \( 1 - 6T + 13T^{2} \)
17 \( 1 + 1.41iT - 17T^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + 7.07T + 23T^{2} \)
29 \( 1 + 8.48iT - 29T^{2} \)
31 \( 1 + 2iT - 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + 4.24iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 - 5.65iT - 53T^{2} \)
59 \( 1 - 11.3T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + 16iT - 67T^{2} \)
71 \( 1 - 4.24T + 71T^{2} \)
73 \( 1 - 14T + 73T^{2} \)
79 \( 1 - 4iT - 79T^{2} \)
83 \( 1 - 8.48T + 83T^{2} \)
89 \( 1 + 7.07iT - 89T^{2} \)
97 \( 1 - 6T + 97T^{2} \)
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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

−10.51098387511703739317886139374, −9.355906069603127123067578867945, −8.539786344947961811886504662070, −7.72142211051777158833357119158, −6.50573491882623869180484053423, −5.95323126076860277069743914398, −4.94589299982810611379148849924, −3.88572930058800319002859268860, −2.12091020995443491702709848618, −0.78275659100519606258338743859, 1.37957040085384193193361657961, 3.46180807729881421563979511984, 4.01493036803378932483330905584, 5.36982684689779848746050659130, 6.28178835773177346930465721014, 6.84176252730388801822811237722, 8.167212773613111909765310596915, 9.056532531878420881218316939245, 10.20639985335889591031378794870, 10.60669516030233059335209844446

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