↑

Properties

Label 2-672-168.125-c3-0-18
Degree $2$
Conductor $672$
Sign $-0.416 - 0.909i$
Analytic cond. $39.6492$
Root an. cond. $6.29676$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (4.05 + 3.24i)3-s − 13.2i·5-s + (−18.4 − 1.29i)7-s + (5.90 + 26.3i)9-s − 49.5·11-s + 28.7·13-s + (42.9 − 53.7i)15-s + 69.2·17-s + 24.7·19-s + (−70.7 − 65.2i)21-s + 85.5i·23-s − 50.2·25-s + (−61.5 + 126. i)27-s − 111.·29-s + 236. i·31-s + ⋯
L(s)  = 1  + (0.780 + 0.624i)3-s − 1.18i·5-s + (−0.997 − 0.0700i)7-s + (0.218 + 0.975i)9-s − 1.35·11-s + 0.614·13-s + (0.740 − 0.924i)15-s + 0.988·17-s + 0.299·19-s + (−0.734 − 0.678i)21-s + 0.775i·23-s − 0.402·25-s + (−0.439 + 0.898i)27-s − 0.714·29-s + 1.37i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.416 - 0.909i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.416 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $-0.416 - 0.909i$
Analytic conductor: \(39.6492\)
Root analytic conductor: \(6.29676\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :3/2),\ -0.416 - 0.909i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.295735917\)
\(L(\frac12)\) \(\approx\) \(1.295735917\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.05 - 3.24i)T \)
7 \( 1 + (18.4 + 1.29i)T \)
good5 \( 1 + 13.2iT - 125T^{2} \)
11 \( 1 + 49.5T + 1.33e3T^{2} \)
13 \( 1 - 28.7T + 2.19e3T^{2} \)
17 \( 1 - 69.2T + 4.91e3T^{2} \)
19 \( 1 - 24.7T + 6.85e3T^{2} \)
23 \( 1 - 85.5iT - 1.21e4T^{2} \)
29 \( 1 + 111.T + 2.43e4T^{2} \)
31 \( 1 - 236. iT - 2.97e4T^{2} \)
37 \( 1 - 261. iT - 5.06e4T^{2} \)
41 \( 1 + 471.T + 6.89e4T^{2} \)
43 \( 1 - 261. iT - 7.95e4T^{2} \)
47 \( 1 + 217.T + 1.03e5T^{2} \)
53 \( 1 + 11.8T + 1.48e5T^{2} \)
59 \( 1 + 236. iT - 2.05e5T^{2} \)
61 \( 1 - 754.T + 2.26e5T^{2} \)
67 \( 1 + 163. iT - 3.00e5T^{2} \)
71 \( 1 - 478. iT - 3.57e5T^{2} \)
73 \( 1 - 1.03e3iT - 3.89e5T^{2} \)
79 \( 1 - 148.T + 4.93e5T^{2} \)
83 \( 1 - 739. iT - 5.71e5T^{2} \)
89 \( 1 + 1.34e3T + 7.04e5T^{2} \)
97 \( 1 + 183. iT - 9.12e5T^{2} \)
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

−9.961528056307829936362808014598, −9.703491245169715193737696122776, −8.547776685886889345656668332691, −8.150880121974722225163497448273, −7.01343587813364621686538127001, −5.50931761369697648257566557883, −4.99331310491625738642237154395, −3.68079796880586125280204328519, −2.93899304589358541127430711914, −1.34694725343688880843320077342, 0.32534742908411260660608640945, 2.18212621944766898497752631455, 3.04447671578467982694640607541, 3.70331780011683430057135046770, 5.57480763507895204875964600272, 6.43992474640188549592203548342, 7.25363957529940780559777958623, 7.907797092816356443028987673116, 8.920663294108462880897965938928, 9.965754784025746385658049052320

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