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Properties

Label 2-672-28.27-c1-0-14
Degree $2$
Conductor $672$
Sign $0.110 + 0.993i$
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  + 3-s − 4.33i·5-s + (1.65 − 2.06i)7-s + 9-s + 3.79i·11-s + 2.82i·13-s − 4.33i·15-s − 4.33i·17-s + 2.54·19-s + (1.65 − 2.06i)21-s − 5.64i·23-s − 13.8·25-s + 27-s − 9.50·29-s + 1.84·31-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.93i·5-s + (0.624 − 0.781i)7-s + 0.333·9-s + 1.14i·11-s + 0.784i·13-s − 1.11i·15-s − 1.05i·17-s + 0.582·19-s + (0.360 − 0.450i)21-s − 1.17i·23-s − 2.76·25-s + 0.192·27-s − 1.76·29-s + 0.331·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.110 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.110 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.110 + 0.993i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ 0.110 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40332 - 1.25569i\)
\(L(\frac12)\) \(\approx\) \(1.40332 - 1.25569i\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 + (-1.65 + 2.06i)T \)
good5 \( 1 + 4.33iT - 5T^{2} \)
11 \( 1 - 3.79iT - 11T^{2} \)
13 \( 1 - 2.82iT - 13T^{2} \)
17 \( 1 + 4.33iT - 17T^{2} \)
19 \( 1 - 2.54T + 19T^{2} \)
23 \( 1 + 5.64iT - 23T^{2} \)
29 \( 1 + 9.50T + 29T^{2} \)
31 \( 1 - 1.84T + 31T^{2} \)
37 \( 1 - 5.11T + 37T^{2} \)
41 \( 1 - 1.32iT - 41T^{2} \)
43 \( 1 - 2.47iT - 43T^{2} \)
47 \( 1 - 12.2T + 47T^{2} \)
53 \( 1 - 1.23T + 53T^{2} \)
59 \( 1 + 1.65T + 59T^{2} \)
61 \( 1 - 3.50iT - 61T^{2} \)
67 \( 1 - 11.1iT - 67T^{2} \)
71 \( 1 - 3.03iT - 71T^{2} \)
73 \( 1 - 3.01iT - 73T^{2} \)
79 \( 1 - 10.1iT - 79T^{2} \)
83 \( 1 - 3.04T + 83T^{2} \)
89 \( 1 + 6.53iT - 89T^{2} \)
97 \( 1 - 6.33iT - 97T^{2} \)
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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

−9.943679702999394105193598795354, −9.364538079897422448913819564818, −8.671465136714486591718994473663, −7.74269418306560031462692340027, −7.11313817774676490590915592709, −5.47144290432419160601602282245, −4.52724933738373377326960140233, −4.16609295625747102122955072515, −2.14217644890542946699619683490, −1.01346280904488804929174107168, 2.03223047119631044791839955611, 3.09006779093623381268103510236, 3.71599298018287165749167569589, 5.61233688978505435011142375639, 6.12955556622028738607023237408, 7.46279230025535519413296693977, 7.88093300472034232489831920016, 8.963771520311157997546648344898, 9.914860973287207148651102289923, 10.90957603949486434039975532537

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