Properties

Degree $2$
Conductor $210$
Sign $-0.487 + 0.872i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·2-s + (0.618 − 1.61i)3-s − 4-s + 5-s + (−1.61 − 0.618i)6-s + (0.381 − 2.61i)7-s + i·8-s + (−2.23 − 2.00i)9-s i·10-s + 4.47i·11-s + (−0.618 + 1.61i)12-s − 3.23i·13-s + (−2.61 − 0.381i)14-s + (0.618 − 1.61i)15-s + 16-s + 0.763·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.356 − 0.934i)3-s − 0.5·4-s + 0.447·5-s + (−0.660 − 0.252i)6-s + (0.144 − 0.989i)7-s + 0.353i·8-s + (−0.745 − 0.666i)9-s − 0.316i·10-s + 1.34i·11-s + (−0.178 + 0.467i)12-s − 0.897i·13-s + (−0.699 − 0.102i)14-s + (0.159 − 0.417i)15-s + 0.250·16-s + 0.185·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.487 + 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.487 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.487 + 0.872i$
Motivic weight: \(1\)
Character: $\chi_{210} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ -0.487 + 0.872i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.670821 - 1.14352i\)
\(L(\frac12)\) \(\approx\) \(0.670821 - 1.14352i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (-0.618 + 1.61i)T \)
5 \( 1 - T \)
7 \( 1 + (-0.381 + 2.61i)T \)
good11 \( 1 - 4.47iT - 11T^{2} \)
13 \( 1 + 3.23iT - 13T^{2} \)
17 \( 1 - 0.763T + 17T^{2} \)
19 \( 1 - 0.472iT - 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 + 5.70iT - 29T^{2} \)
31 \( 1 - 7.23iT - 31T^{2} \)
37 \( 1 - 5.23T + 37T^{2} \)
41 \( 1 - 6.47T + 41T^{2} \)
43 \( 1 - 12.9T + 43T^{2} \)
47 \( 1 - 2.47T + 47T^{2} \)
53 \( 1 - 8.47iT - 53T^{2} \)
59 \( 1 + 4.47T + 59T^{2} \)
61 \( 1 + 2.76iT - 61T^{2} \)
67 \( 1 + 12T + 67T^{2} \)
71 \( 1 - 2.76iT - 71T^{2} \)
73 \( 1 - 6.76iT - 73T^{2} \)
79 \( 1 - 8.94T + 79T^{2} \)
83 \( 1 + 16.6T + 83T^{2} \)
89 \( 1 + 14.4T + 89T^{2} \)
97 \( 1 + 5.23iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.32780765369404975502391892676, −11.09988515350571121773481294425, −10.11999641535239749876470025559, −9.274618619610422215752940428211, −7.88102971389047702579319886662, −7.21880171673753600146579085201, −5.78020523149887817724904724686, −4.25110150697653622959183206661, −2.72901789556295232313364175973, −1.29525487535678039644420450590, 2.70748439083832360578983353683, 4.24058733389538653244322784739, 5.49291523962495038756578242293, 6.22383389065855848885223247386, 7.919321663811144579775381784087, 8.938106051143470080906408762661, 9.312504598044199026391606113011, 10.66756847346795891634953659027, 11.56052753987791241564289996984, 12.90665855516648740145117309923

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