Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $-0.999 - 0.0260i$
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 + (0.282 − 2.98i)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 + (5.97 + 0.565i)12-s + (2.90 − 2.90i)13-s + (9.63 + 2.28i)14-s + (−10.3 − 10.8i)15-s − 4·16-s + (−16.3 + 16.3i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (0.0942 − 0.995i)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 + (0.497 + 0.0471i)12-s + (0.223 − 0.223i)13-s + (0.688 + 0.162i)14-s + (−0.688 − 0.724i)15-s − 0.250·16-s + (−0.959 + 0.959i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0260i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 - 0.0260i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.999 - 0.0260i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (83, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ -0.999 - 0.0260i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.0115803 + 0.887580i\)
\(L(\frac12)\)  \(\approx\)  \(0.0115803 + 0.887580i\)
\(L(2)\)   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 \( 1 + (-0.282 + 2.98i)T \)
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

−11.70766116928327619737237982192, −10.81004389879611133287532895118, −9.386602042533244057238627572843, −8.771270271798546449243162740264, −7.931052975016401040320915730900, −6.30231888489774539936148132993, −5.72995532731998971045755211926, −3.46972209870836769276675153039, −2.11790772074459160935289752183, −0.55477409240133610815876452989, 2.45457829318071739745509379000, 4.07396816993206285754733110332, 5.36549024891683794583686136622, 6.68691536334699946390342780565, 7.36124821626791826661932583491, 9.131228148043325982628768222036, 9.633439176584079434801888259683, 10.36596323028972703610078043692, 11.21294667798115275058191883442, 12.72605858526855648055581271049

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