↑

Properties

Label 2-672-21.20-c1-0-31
Degree $2$
Conductor $672$
Sign $-0.966 - 0.255i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.848 − 1.51i)3-s − 1.69·5-s + (−2.56 + 0.662i)7-s + (−1.56 − 2.56i)9-s − 4i·11-s + 4.34i·13-s + (−1.43 + 2.56i)15-s − 6.04·17-s + 3.02i·19-s + (−1.17 + 4.43i)21-s − 1.12i·23-s − 2.12·25-s + (−5.19 + 0.185i)27-s + 1.12i·29-s − 4.71i·31-s + ⋯
L(s)  = 1  + (0.489 − 0.871i)3-s − 0.758·5-s + (−0.968 + 0.250i)7-s + (−0.520 − 0.853i)9-s − 1.20i·11-s + 1.20i·13-s + (−0.371 + 0.661i)15-s − 1.46·17-s + 0.692i·19-s + (−0.255 + 0.966i)21-s − 0.234i·23-s − 0.424·25-s + (−0.999 + 0.0357i)27-s + 0.208i·29-s − 0.847i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 - 0.255i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.966 - 0.255i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $-0.966 - 0.255i$
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: \((2,\ 672,\ (\ :1/2),\ -0.966 - 0.255i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0428936 + 0.329744i\)
\(L(\frac12)\) \(\approx\) \(0.0428936 + 0.329744i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.848 + 1.51i)T \)
7 \( 1 + (2.56 - 0.662i)T \)
good5 \( 1 + 1.69T + 5T^{2} \)
11 \( 1 + 4iT - 11T^{2} \)
13 \( 1 - 4.34iT - 13T^{2} \)
17 \( 1 + 6.04T + 17T^{2} \)
19 \( 1 - 3.02iT - 19T^{2} \)
23 \( 1 + 1.12iT - 23T^{2} \)
29 \( 1 - 1.12iT - 29T^{2} \)
31 \( 1 + 4.71iT - 31T^{2} \)
37 \( 1 + 8.24T + 37T^{2} \)
41 \( 1 + 6.04T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + 3.39T + 47T^{2} \)
53 \( 1 + 9.12iT - 53T^{2} \)
59 \( 1 - 10.3T + 59T^{2} \)
61 \( 1 - 11.1iT - 61T^{2} \)
67 \( 1 - 2.24T + 67T^{2} \)
71 \( 1 + 14.2iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 5.12T + 79T^{2} \)
83 \( 1 + 17.1T + 83T^{2} \)
89 \( 1 - 9.43T + 89T^{2} \)
97 \( 1 + 15.4iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.881645458039419225612149062602, −8.802550709428995093719280074069, −8.527099739897391758406729951756, −7.30018356195805915394077014970, −6.60140343915208261694290413333, −5.83679982456436860255537726014, −4.10921504062110761520382640404, −3.30400448619765335452196114913, −2.06106508623185349797189227459, −0.15458845594749216891372834598, 2.47754371457772736704462385591, 3.53544647073113482162883013273, 4.35995884517096078722091756071, 5.31069504475637572541903567221, 6.73401316453449463251400552363, 7.51295911010518918955769088064, 8.505666319128472644636642469452, 9.299855544018280577793255841328, 10.12874919433803700727561758663, 10.71700880283905249482028634192

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