Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $-0.905 - 0.424i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + i)2-s + (−0.282 − 2.98i)3-s − 2i·4-s + (−3.28 − 3.77i)5-s + (3.26 + 2.70i)6-s + (−3.67 − 5.95i)7-s + (2 + 2i)8-s + (−8.84 + 1.68i)9-s + (7.05 + 0.487i)10-s + 19.5i·11-s + (−5.97 + 0.565i)12-s + (−2.90 − 2.90i)13-s + (9.63 + 2.28i)14-s + (−10.3 + 10.8i)15-s − 4·16-s + (16.3 + 16.3i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (−0.0942 − 0.995i)3-s − 0.5i·4-s + (−0.656 − 0.754i)5-s + (0.544 + 0.450i)6-s + (−0.525 − 0.850i)7-s + (0.250 + 0.250i)8-s + (−0.982 + 0.187i)9-s + (0.705 + 0.0487i)10-s + 1.77i·11-s + (−0.497 + 0.0471i)12-s + (−0.223 − 0.223i)13-s + (0.688 + 0.162i)14-s + (−0.688 + 0.724i)15-s − 0.250·16-s + (0.959 + 0.959i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.905 - 0.424i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.905 - 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.905 - 0.424i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (167, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ -0.905 - 0.424i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.0266472 + 0.119498i\)
\(L(\frac12)\)  \(\approx\)  \(0.0266472 + 0.119498i\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (1 - i)T \)
3 \( 1 + (0.282 + 2.98i)T \)
5 \( 1 + (3.28 + 3.77i)T \)
7 \( 1 + (3.67 + 5.95i)T \)
good11 \( 1 - 19.5iT - 121T^{2} \)
13 \( 1 + (2.90 + 2.90i)T + 169iT^{2} \)
17 \( 1 + (-16.3 - 16.3i)T + 289iT^{2} \)
19 \( 1 + 8.66T + 361T^{2} \)
23 \( 1 + (6.73 + 6.73i)T + 529iT^{2} \)
29 \( 1 + 31.3T + 841T^{2} \)
31 \( 1 + 39.4iT - 961T^{2} \)
37 \( 1 + (25.1 + 25.1i)T + 1.36e3iT^{2} \)
41 \( 1 + 58.9T + 1.68e3T^{2} \)
43 \( 1 + (-10.5 + 10.5i)T - 1.84e3iT^{2} \)
47 \( 1 + (29.2 + 29.2i)T + 2.20e3iT^{2} \)
53 \( 1 + (10.3 + 10.3i)T + 2.80e3iT^{2} \)
59 \( 1 + 42.5iT - 3.48e3T^{2} \)
61 \( 1 - 45.1iT - 3.72e3T^{2} \)
67 \( 1 + (-89.3 - 89.3i)T + 4.48e3iT^{2} \)
71 \( 1 - 47.3iT - 5.04e3T^{2} \)
73 \( 1 + (89.3 + 89.3i)T + 5.32e3iT^{2} \)
79 \( 1 - 41.4iT - 6.24e3T^{2} \)
83 \( 1 + (-44.9 + 44.9i)T - 6.88e3iT^{2} \)
89 \( 1 - 4.80iT - 7.92e3T^{2} \)
97 \( 1 + (2.01 - 2.01i)T - 9.40e3iT^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−11.81044561696431870588304807466, −10.45706515565085305083065358051, −9.526970707733231491675168794635, −8.234969343914911072378228082819, −7.52386810972042783268270752842, −6.79835673046112983463307759421, −5.42797431257327937492730832433, −4.01048147679021818151553206336, −1.70797408133247536581158411397, −0.079973312188955936510472392629, 2.97190781841236906765952077015, 3.53480897687258985540497963497, 5.30373539737611660701303783675, 6.51114896329002528736795958747, 8.064995526637579948550625071452, 8.902862348914959203497503476192, 9.836342507817435858850761907960, 10.79477113144304650548150686823, 11.52386239321719603926406537687, 12.17586323588407403296547358487

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