Properties

Label 2-672-21.20-c1-0-12
Degree $2$
Conductor $672$
Sign $0.936 - 0.350i$
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 + (−1.60 − 4.29i)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 + (−0.350 − 0.936i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 - 0.350i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 - 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.936 - 0.350i$
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: \((2,\ 672,\ (\ :1/2),\ 0.936 - 0.350i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49931 + 0.270964i\)
\(L(\frac12)\) \(\approx\) \(1.49931 + 0.270964i\)
\(L(\frac{3}{2})\) 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

−10.72147787706537242614215890113, −9.584036900790060210459956506818, −9.102798883041358087893421729278, −7.918472278322398995180259129875, −6.67114044515813026207283068064, −5.89780421669818729535245693527, −5.50325963451225259257542351512, −4.30436387511883994361305225292, −2.39070189515421328305050853046, −1.45358080890331479416041109268, 1.07056919577681234930869176654, 2.43917774901478156456105955391, 4.27797895483756676419784234497, 5.04485547176643110540121209345, 5.87056321413901905380938843645, 6.77444026724640094778505286184, 7.61294085792463192706239509141, 8.983242873939117088224197356997, 9.915550861898781575039734133819, 10.48054008224796769189291743017

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