Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $-0.162 + 0.986i$
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 + (−5.95 − 3.67i)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 + (−0.850 − 0.525i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.162 + 0.986i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.162 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.162 + 0.986i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (167, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ -0.162 + 0.986i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.46911 - 1.72998i\)
\(L(\frac12)\)  \(\approx\)  \(1.46911 - 1.72998i\)
\(L(2)\)   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 \( 1 + (5.95 + 3.67i)T \)
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

−12.10999145400224677738355793490, −10.88947929734006511013770881609, −9.961560428487699912990219170524, −8.820033245840786849660207677087, −8.151382996492469899452436331920, −6.73931715862981527690794446377, −5.27932851129706094693628933911, −3.71084159958903031804470472109, −3.32775483678584487143443974561, −1.08012958994623367827963258005, 2.58306640059823980345960275386, 3.51986009268496016489258637910, 4.82489664994871791257367738821, 6.57385414014002151836761847401, 7.27924880769868307467177685125, 8.134172677490020485100930419939, 9.480527052658941347299095104842, 10.15294912273396945599415778544, 11.90737126885627267871332085914, 12.44685446225793795429434828620

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