Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $-0.587 + 0.809i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − i)2-s + (−2.98 − 0.282i)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 + (−0.565 + 5.97i)12-s + (−2.90 − 2.90i)13-s + (−9.63 − 2.28i)14-s + (−8.74 − 12.1i)15-s − 4·16-s + (−16.3 − 16.3i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (−0.995 − 0.0942i)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 + (−0.0471 + 0.497i)12-s + (−0.223 − 0.223i)13-s + (−0.688 − 0.162i)14-s + (−0.582 − 0.812i)15-s − 0.250·16-s + (−0.959 − 0.959i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.587 + 0.809i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.587 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.587 + 0.809i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (167, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ -0.587 + 0.809i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.558538 - 1.09526i\)
\(L(\frac12)\)  \(\approx\)  \(0.558538 - 1.09526i\)
\(L(2)\)   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 \( 1 + (2.98 + 0.282i)T \)
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 \( 1 + (2.01 - 2.01i)T - 9.40e3iT^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−11.54291805714622826255573933484, −10.91832590749243142747828890863, −10.31466342106207210585303456744, −9.205165670275871895939605253489, −7.31440301042805742727148716689, −6.35354964297605208194838560334, −5.62733791623498617165528775997, −4.15270147461297126304212263111, −2.75400150311778457219234467426, −0.65570629848420308370985754999, 2.05173488731542428434809544692, 4.39694153175102520068161980177, 5.09495523104142252963896438216, 6.24664697345056652268412687258, 6.91564077337082676651490101086, 8.575808512321656403153549046925, 9.569074006010151681703177918241, 10.47257598592159628069160702901, 11.95510093919304171409684833352, 12.64167417598791093464550142665

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