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 + (−1.61 − 0.618i)3-s − 4-s + 5-s + (0.618 − 1.61i)6-s + (2.61 + 0.381i)7-s i·8-s + (2.23 + 2.00i)9-s + i·10-s + 4.47i·11-s + (1.61 + 0.618i)12-s − 1.23i·13-s + (−0.381 + 2.61i)14-s + (−1.61 − 0.618i)15-s + 16-s + 5.23·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.934 − 0.356i)3-s − 0.5·4-s + 0.447·5-s + (0.252 − 0.660i)6-s + (0.989 + 0.144i)7-s − 0.353i·8-s + (0.745 + 0.666i)9-s + 0.316i·10-s + 1.34i·11-s + (0.467 + 0.178i)12-s − 0.342i·13-s + (−0.102 + 0.699i)14-s + (−0.417 − 0.159i)15-s + 0.250·16-s + 1.26·17-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

\(L(1)\) \(\approx\) \(0.885203 + 0.519284i\)
\(L(\frac12)\) \(\approx\) \(0.885203 + 0.519284i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.61 + 0.618i)T \)
5 \( 1 - T \)
7 \( 1 + (-2.61 - 0.381i)T \)
good11 \( 1 - 4.47iT - 11T^{2} \)
13 \( 1 + 1.23iT - 13T^{2} \)
17 \( 1 - 5.23T + 17T^{2} \)
19 \( 1 - 8.47iT - 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 + 7.70iT - 29T^{2} \)
31 \( 1 + 2.76iT - 31T^{2} \)
37 \( 1 - 0.763T + 37T^{2} \)
41 \( 1 + 2.47T + 41T^{2} \)
43 \( 1 + 4.94T + 43T^{2} \)
47 \( 1 + 6.47T + 47T^{2} \)
53 \( 1 - 0.472iT - 53T^{2} \)
59 \( 1 - 4.47T + 59T^{2} \)
61 \( 1 - 7.23iT - 61T^{2} \)
67 \( 1 + 12T + 67T^{2} \)
71 \( 1 + 7.23iT - 71T^{2} \)
73 \( 1 + 11.2iT - 73T^{2} \)
79 \( 1 + 8.94T + 79T^{2} \)
83 \( 1 - 14.6T + 83T^{2} \)
89 \( 1 + 5.52T + 89T^{2} \)
97 \( 1 - 0.763iT - 97T^{2} \)
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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

−12.38597062050940731012397942374, −11.87415649155267587468497862383, −10.37728005729049153696305905483, −9.842717578474260972633046809822, −8.111410762312666473310373948238, −7.52206610716490829259471236966, −6.21026672455125969005904546329, −5.39378971821176985731350793601, −4.37719302623411572614881697721, −1.70144517042571448310252289478, 1.21527653637586685339144657534, 3.33016707163840279440937490097, 4.85894466694509251136112359601, 5.56250779248751694961822887916, 6.98616253069062895879214867105, 8.494393243429295623907305615873, 9.478362294525609883146800929672, 10.57881849207187713242554907023, 11.24251325779676715312717541218, 11.83055703838193839995996042041

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