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Properties

Label 2-672-7.6-c2-0-26
Degree $2$
Conductor $672$
Sign $-0.932 + 0.361i$
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 − 5.15i·5-s + (2.53 + 6.52i)7-s − 2.99·9-s − 14.9·11-s + 4.05i·13-s + 8.92·15-s − 26.1i·17-s + 10.3i·19-s + (−11.3 + 4.38i)21-s − 18.0·23-s − 1.53·25-s − 5.19i·27-s − 42.4·29-s − 3.16i·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 1.03i·5-s + (0.361 + 0.932i)7-s − 0.333·9-s − 1.35·11-s + 0.311i·13-s + 0.594·15-s − 1.54i·17-s + 0.542i·19-s + (−0.538 + 0.208i)21-s − 0.783·23-s − 0.0613·25-s − 0.192i·27-s − 1.46·29-s − 0.102i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.932 + 0.361i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.932 + 0.361i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $-0.932 + 0.361i$
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: \((2,\ 672,\ (\ :1),\ -0.932 + 0.361i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.09026461038\)
\(L(\frac12)\) \(\approx\) \(0.09026461038\)
\(L(2)\) 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 \( 1 + (-2.53 - 6.52i)T \)
good5 \( 1 + 5.15iT - 25T^{2} \)
11 \( 1 + 14.9T + 121T^{2} \)
13 \( 1 - 4.05iT - 169T^{2} \)
17 \( 1 + 26.1iT - 289T^{2} \)
19 \( 1 - 10.3iT - 361T^{2} \)
23 \( 1 + 18.0T + 529T^{2} \)
29 \( 1 + 42.4T + 841T^{2} \)
31 \( 1 + 3.16iT - 961T^{2} \)
37 \( 1 + 34.4T + 1.36e3T^{2} \)
41 \( 1 - 41.4iT - 1.68e3T^{2} \)
43 \( 1 - 16.2T + 1.84e3T^{2} \)
47 \( 1 + 1.01iT - 2.20e3T^{2} \)
53 \( 1 + 84.3T + 2.80e3T^{2} \)
59 \( 1 + 59.3iT - 3.48e3T^{2} \)
61 \( 1 + 86.6iT - 3.72e3T^{2} \)
67 \( 1 + 116.T + 4.48e3T^{2} \)
71 \( 1 + 97.7T + 5.04e3T^{2} \)
73 \( 1 + 28.2iT - 5.32e3T^{2} \)
79 \( 1 - 103.T + 6.24e3T^{2} \)
83 \( 1 + 34.8iT - 6.88e3T^{2} \)
89 \( 1 - 64.9iT - 7.92e3T^{2} \)
97 \( 1 + 105. iT - 9.40e3T^{2} \)
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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

−9.682393035241908231850555625538, −9.175428647695052113060326461665, −8.272092980782967017378476508563, −7.57696659018529578214954804386, −5.99431740189062791006101000826, −5.14216583617678268883563975588, −4.68538241167110484019576151568, −3.16389712020790169813941559568, −1.93802776474140494843230399200, −0.03001314571861373213040568425, 1.77027858922916614606521421374, 2.95880855787517036938349015044, 4.04530372016948067254743804362, 5.41617956941981753416693575151, 6.33884691340161216542771430508, 7.38761126188508026807262708481, 7.73036744213411089075407479257, 8.775018399750164171015561644505, 10.30674024575859820643847933284, 10.54953400401195620637731188672

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