Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $-0.958 - 0.285i$
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 + (−0.947 − 2.84i)3-s + 2i·4-s + (−4.64 − 1.84i)5-s + (1.89 − 3.79i)6-s + (−3.13 + 6.25i)7-s + (−2 + 2i)8-s + (−7.20 + 5.39i)9-s + (−2.79 − 6.49i)10-s − 2.08i·11-s + (5.69 − 1.89i)12-s + (−8.39 + 8.39i)13-s + (−9.39 + 3.12i)14-s + (−0.854 + 14.9i)15-s − 4·16-s + (−4.96 + 4.96i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (−0.315 − 0.948i)3-s + 0.5i·4-s + (−0.929 − 0.369i)5-s + (0.316 − 0.632i)6-s + (−0.448 + 0.893i)7-s + (−0.250 + 0.250i)8-s + (−0.800 + 0.599i)9-s + (−0.279 − 0.649i)10-s − 0.189i·11-s + (0.474 − 0.157i)12-s + (−0.645 + 0.645i)13-s + (−0.671 + 0.222i)14-s + (−0.0569 + 0.998i)15-s − 0.250·16-s + (−0.292 + 0.292i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.958 - 0.285i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.958 - 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.958 - 0.285i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (83, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ -0.958 - 0.285i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.0421707 + 0.288930i\)
\(L(\frac12)\)  \(\approx\)  \(0.0421707 + 0.288930i\)
\(L(2)\)   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 \( 1 + (0.947 + 2.84i)T \)
5 \( 1 + (4.64 + 1.84i)T \)
7 \( 1 + (3.13 - 6.25i)T \)
good11 \( 1 + 2.08iT - 121T^{2} \)
13 \( 1 + (8.39 - 8.39i)T - 169iT^{2} \)
17 \( 1 + (4.96 - 4.96i)T - 289iT^{2} \)
19 \( 1 + 17.3T + 361T^{2} \)
23 \( 1 + (3.08 - 3.08i)T - 529iT^{2} \)
29 \( 1 + 39.1T + 841T^{2} \)
31 \( 1 + 42.3iT - 961T^{2} \)
37 \( 1 + (-36.7 + 36.7i)T - 1.36e3iT^{2} \)
41 \( 1 + 15.5T + 1.68e3T^{2} \)
43 \( 1 + (-22.8 - 22.8i)T + 1.84e3iT^{2} \)
47 \( 1 + (33.4 - 33.4i)T - 2.20e3iT^{2} \)
53 \( 1 + (-59.7 + 59.7i)T - 2.80e3iT^{2} \)
59 \( 1 - 48.9iT - 3.48e3T^{2} \)
61 \( 1 + 82.9iT - 3.72e3T^{2} \)
67 \( 1 + (54.8 - 54.8i)T - 4.48e3iT^{2} \)
71 \( 1 - 74.9iT - 5.04e3T^{2} \)
73 \( 1 + (75.1 - 75.1i)T - 5.32e3iT^{2} \)
79 \( 1 - 3.61iT - 6.24e3T^{2} \)
83 \( 1 + (-103. - 103. i)T + 6.88e3iT^{2} \)
89 \( 1 - 24.4iT - 7.92e3T^{2} \)
97 \( 1 + (-35.3 - 35.3i)T + 9.40e3iT^{2} \)
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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

−12.76454598346252339370955524388, −11.79434804272452701710349666215, −11.21782186616263238287444682352, −9.288760676757068004983211875372, −8.320958561120614273389164026340, −7.43981516166171375067143736992, −6.41010474446613538123523249666, −5.44361526598608483528805526683, −4.07939868945857531386591902028, −2.37514281839438226126029386476, 0.13614703011569606050213431824, 3.03087768411972251034015642307, 4.01390437820734297960290855818, 4.89758514946695973806355201046, 6.38503040219079131527190046286, 7.53296781028373893090436368412, 8.978872382513301103348733601573, 10.23689467759491900514677305960, 10.63387251338002053217363080947, 11.62136141985261181836748356703

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