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Properties

Label 2-672-7.6-c2-0-14
Degree $2$
Conductor $672$
Sign $0.378 - 0.925i$
Analytic cond. $18.3106$
Root an. cond. $4.27909$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73i·3-s − 0.817i·5-s + (6.47 + 2.64i)7-s − 2.99·9-s + 7.22·11-s + 5.77i·13-s + 1.41·15-s + 15.3i·17-s − 17.6i·19-s + (−4.58 + 11.2i)21-s − 8.69·23-s + 24.3·25-s − 5.19i·27-s + 19.4·29-s + 0.274i·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 0.163i·5-s + (0.925 + 0.378i)7-s − 0.333·9-s + 0.656·11-s + 0.443i·13-s + 0.0944·15-s + 0.903i·17-s − 0.931i·19-s + (−0.218 + 0.534i)21-s − 0.378·23-s + 0.973·25-s − 0.192i·27-s + 0.671·29-s + 0.00884i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.378 - 0.925i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.378 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.378 - 0.925i$
Analytic conductor: \(18.3106\)
Root analytic conductor: \(4.27909\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1),\ 0.378 - 0.925i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.026274383\)
\(L(\frac12)\) \(\approx\) \(2.026274383\)
\(L(2)\) 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 \( 1 - 1.73iT \)
7 \( 1 + (-6.47 - 2.64i)T \)
good5 \( 1 + 0.817iT - 25T^{2} \)
11 \( 1 - 7.22T + 121T^{2} \)
13 \( 1 - 5.77iT - 169T^{2} \)
17 \( 1 - 15.3iT - 289T^{2} \)
19 \( 1 + 17.6iT - 361T^{2} \)
23 \( 1 + 8.69T + 529T^{2} \)
29 \( 1 - 19.4T + 841T^{2} \)
31 \( 1 - 0.274iT - 961T^{2} \)
37 \( 1 + 4.34T + 1.36e3T^{2} \)
41 \( 1 - 72.6iT - 1.68e3T^{2} \)
43 \( 1 - 29.3T + 1.84e3T^{2} \)
47 \( 1 - 15.7iT - 2.20e3T^{2} \)
53 \( 1 + 18.2T + 2.80e3T^{2} \)
59 \( 1 - 51.3iT - 3.48e3T^{2} \)
61 \( 1 - 21.5iT - 3.72e3T^{2} \)
67 \( 1 - 45.5T + 4.48e3T^{2} \)
71 \( 1 + 100.T + 5.04e3T^{2} \)
73 \( 1 - 117. iT - 5.32e3T^{2} \)
79 \( 1 - 61.7T + 6.24e3T^{2} \)
83 \( 1 + 65.3iT - 6.88e3T^{2} \)
89 \( 1 - 105. iT - 7.92e3T^{2} \)
97 \( 1 - 59.9iT - 9.40e3T^{2} \)
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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

−10.55251615085540800919151901524, −9.498491400349084559612526773525, −8.767238102069740859780626080071, −8.105929574553171488437638855752, −6.86785639917364204499858869027, −5.89401299849216754976514607791, −4.80165687107467878056603306590, −4.16359652633601274640681364727, −2.74114403591550903008453276895, −1.33919739151121474248956027436, 0.837048550105250535117897982409, 2.06360824099539904051374171649, 3.44216603881005648238258218049, 4.64584612014847873856378039841, 5.64820254596566695295997124726, 6.72850131661735661180955823610, 7.50566580814986572185004993253, 8.281726929638799049345003613001, 9.156699748379115239197137266213, 10.29970642309128060004929630968

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