Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $-0.641 + 0.766i$
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 + (1.94 − 2.28i)3-s − 2i·4-s + (−4.58 − 1.99i)5-s + (−0.347 − 4.22i)6-s + (5.57 − 4.23i)7-s + (−2 − 2i)8-s + (−1.46 − 8.87i)9-s + (−6.58 + 2.58i)10-s + 14.6i·11-s + (−4.57 − 3.88i)12-s + (−3.48 − 3.48i)13-s + (1.34 − 9.80i)14-s + (−13.4 + 6.61i)15-s − 4·16-s + (−20.1 − 20.1i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (0.646 − 0.762i)3-s − 0.5i·4-s + (−0.916 − 0.399i)5-s + (−0.0579 − 0.704i)6-s + (0.796 − 0.604i)7-s + (−0.250 − 0.250i)8-s + (−0.163 − 0.986i)9-s + (−0.658 + 0.258i)10-s + 1.32i·11-s + (−0.381 − 0.323i)12-s + (−0.267 − 0.267i)13-s + (0.0961 − 0.700i)14-s + (−0.897 + 0.441i)15-s − 0.250·16-s + (−1.18 − 1.18i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.641 + 0.766i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.641 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.641 + 0.766i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (167, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ -0.641 + 0.766i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.892632 - 1.91075i\)
\(L(\frac12)\)  \(\approx\)  \(0.892632 - 1.91075i\)
\(L(2)\)   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 \( 1 + (-1.94 + 2.28i)T \)
5 \( 1 + (4.58 + 1.99i)T \)
7 \( 1 + (-5.57 + 4.23i)T \)
good11 \( 1 - 14.6iT - 121T^{2} \)
13 \( 1 + (3.48 + 3.48i)T + 169iT^{2} \)
17 \( 1 + (20.1 + 20.1i)T + 289iT^{2} \)
19 \( 1 - 26.4T + 361T^{2} \)
23 \( 1 + (-2.68 - 2.68i)T + 529iT^{2} \)
29 \( 1 - 28.5T + 841T^{2} \)
31 \( 1 - 15.6iT - 961T^{2} \)
37 \( 1 + (-7.69 - 7.69i)T + 1.36e3iT^{2} \)
41 \( 1 - 37.9T + 1.68e3T^{2} \)
43 \( 1 + (-41.7 + 41.7i)T - 1.84e3iT^{2} \)
47 \( 1 + (-21.0 - 21.0i)T + 2.20e3iT^{2} \)
53 \( 1 + (47.4 + 47.4i)T + 2.80e3iT^{2} \)
59 \( 1 - 61.6iT - 3.48e3T^{2} \)
61 \( 1 - 54.1iT - 3.72e3T^{2} \)
67 \( 1 + (-68.9 - 68.9i)T + 4.48e3iT^{2} \)
71 \( 1 - 65.9iT - 5.04e3T^{2} \)
73 \( 1 + (-6.51 - 6.51i)T + 5.32e3iT^{2} \)
79 \( 1 + 42.7iT - 6.24e3T^{2} \)
83 \( 1 + (-9.52 + 9.52i)T - 6.88e3iT^{2} \)
89 \( 1 + 19.3iT - 7.92e3T^{2} \)
97 \( 1 + (84.6 - 84.6i)T - 9.40e3iT^{2} \)
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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

−11.94781870868705131133190491700, −11.21224134186268156861921150531, −9.811320202471316278085945206256, −8.762974793850033179376846632451, −7.51265794274953767237827544044, −7.07383019095618609983080599314, −5.02803822087723049121931096934, −4.12983244947209703432004473906, −2.62060972094570919348221790489, −1.02825871201042733956365361506, 2.72980892443720551159826951616, 3.89014255444628830331712113342, 4.89552167495820927962596894449, 6.18766822685251447730127431886, 7.73826954994199319623167140600, 8.330864173709682244100292056001, 9.234206828494135378565345887687, 10.91795833207792233078724120154, 11.31149234075941542853472896991, 12.52350210358745570891720209115

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