↑

Properties

Label 2-672-168.125-c1-0-14
Degree $2$
Conductor $672$
Sign $0.970 + 0.242i$
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  + (−0.420 + 1.68i)3-s − 2.16i·5-s − 2.64·7-s + (−2.64 − 1.41i)9-s + 5.59·13-s + (3.64 + 0.913i)15-s + 8.66·19-s + (1.11 − 4.44i)21-s − 7.48i·23-s + 0.291·25-s + (3.48 − 3.85i)27-s + 5.74i·35-s + (−2.35 + 9.39i)39-s + (−3.06 + 5.74i)45-s + 7.00·49-s + ⋯
L(s)  = 1  + (−0.242 + 0.970i)3-s − 0.970i·5-s − 0.999·7-s + (−0.881 − 0.471i)9-s + 1.55·13-s + (0.941 + 0.235i)15-s + 1.98·19-s + (0.242 − 0.970i)21-s − 1.56i·23-s + 0.0583·25-s + (0.671 − 0.740i)27-s + 0.970i·35-s + (−0.376 + 1.50i)39-s + (−0.457 + 0.855i)45-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.970 + 0.242i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ 0.970 + 0.242i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23971 - 0.152907i\)
\(L(\frac12)\) \(\approx\) \(1.23971 - 0.152907i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.420 - 1.68i)T \)
7 \( 1 + 2.64T \)
good5 \( 1 + 2.16iT - 5T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 5.59T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 8.66T + 19T^{2} \)
23 \( 1 + 7.48iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 14.4iT - 59T^{2} \)
61 \( 1 - 0.543T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 5.65iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 5.29T + 79T^{2} \)
83 \( 1 + 1.40iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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

−10.37247006434419575359490694975, −9.543887866465113864706144684691, −8.936851494970380671219015307579, −8.191371382558563327523478389466, −6.69469525999849229497751540882, −5.82475173046223939300634851967, −4.98662713869548677817574083190, −3.92473226441166901078089544784, −3.06037379985377977031956971480, −0.834526436888523029814040677902, 1.25076174708443541738822787904, 2.91265459068667039184520673851, 3.55636502717339185873736399963, 5.52576165232138331833419148461, 6.14971077563122944773862122178, 7.04307089284189348413024206859, 7.59679058895968622383680509080, 8.789446316197091024707889935765, 9.708147135164625111671581363620, 10.69804963022587134616566264745

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