↑

Properties

Label 2-672-224.171-c1-0-30
Degree $2$
Conductor $672$
Sign $0.994 - 0.102i$
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  + (−0.630 + 1.26i)2-s + (0.130 − 0.991i)3-s + (−1.20 − 1.59i)4-s + (0.0526 + 0.400i)5-s + (1.17 + 0.790i)6-s + (2.10 − 1.60i)7-s + (2.78 − 0.518i)8-s + (−0.965 − 0.258i)9-s + (−0.539 − 0.185i)10-s + (2.92 + 3.80i)11-s + (−1.73 + 0.986i)12-s + (−6.09 + 2.52i)13-s + (0.701 + 3.67i)14-s + 0.403·15-s + (−1.09 + 3.84i)16-s + (2.33 − 4.05i)17-s + ⋯
L(s)  = 1  + (−0.445 + 0.895i)2-s + (0.0753 − 0.572i)3-s + (−0.602 − 0.798i)4-s + (0.0235 + 0.179i)5-s + (0.478 + 0.322i)6-s + (0.795 − 0.605i)7-s + (0.983 − 0.183i)8-s + (−0.321 − 0.0862i)9-s + (−0.170 − 0.0587i)10-s + (0.881 + 1.14i)11-s + (−0.502 + 0.284i)12-s + (−1.68 + 0.699i)13-s + (0.187 + 0.982i)14-s + 0.104·15-s + (−0.274 + 0.961i)16-s + (0.567 − 0.982i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.102i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.994 - 0.102i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (619, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ 0.994 - 0.102i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26734 + 0.0653936i\)
\(L(\frac12)\) \(\approx\) \(1.26734 + 0.0653936i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.630 - 1.26i)T \)
3 \( 1 + (-0.130 + 0.991i)T \)
7 \( 1 + (-2.10 + 1.60i)T \)
good5 \( 1 + (-0.0526 - 0.400i)T + (-4.82 + 1.29i)T^{2} \)
11 \( 1 + (-2.92 - 3.80i)T + (-2.84 + 10.6i)T^{2} \)
13 \( 1 + (6.09 - 2.52i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 + (-2.33 + 4.05i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.66 + 3.47i)T + (-4.91 - 18.3i)T^{2} \)
23 \( 1 + (-0.913 + 3.40i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-7.84 + 3.25i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (-1.42 + 2.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.145 + 0.0191i)T + (35.7 - 9.57i)T^{2} \)
41 \( 1 + (-2.76 - 2.76i)T + 41iT^{2} \)
43 \( 1 + (3.32 - 8.03i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + (4.10 - 2.36i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-9.48 + 7.27i)T + (13.7 - 51.1i)T^{2} \)
59 \( 1 + (-3.01 - 3.92i)T + (-15.2 + 56.9i)T^{2} \)
61 \( 1 + (-1.33 + 1.73i)T + (-15.7 - 58.9i)T^{2} \)
67 \( 1 + (-0.542 - 0.0714i)T + (64.7 + 17.3i)T^{2} \)
71 \( 1 + (1.15 - 1.15i)T - 71iT^{2} \)
73 \( 1 + (3.75 - 1.00i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (6.24 + 10.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.36 + 3.30i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (-2.02 - 0.541i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + 10.5iT - 97T^{2} \)
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

−10.11217116433778386076251689680, −9.649473788260646314385305909743, −8.661226819778516866456909814492, −7.58143349505301536367220945521, −7.12905665508776628387746610283, −6.51782850517048982529327435698, −4.84858620811360138910079748854, −4.60660479106607844061165481148, −2.43389757760314348721727461598, −0.988399883635022162827255770284, 1.25510916223992386576771753478, 2.76020375182145167964433442226, 3.67941332947426119524815531016, 4.91093887190316365382252178130, 5.61281817020027961369812680848, 7.29554395964146847632589306815, 8.367641457156113329852426533327, 8.731543317643297625811379649807, 9.788555049335616094396560758979, 10.41063122658096969800975940877

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