Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $0.688 - 0.725i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.36 − 0.366i)2-s + (−2.49 + 1.66i)3-s + (1.73 + i)4-s + (−4.84 − 1.24i)5-s + (4.01 − 1.35i)6-s + (−0.400 − 6.98i)7-s + (−1.99 − 2i)8-s + (3.46 − 8.30i)9-s + (6.15 + 3.47i)10-s + (4.21 + 2.43i)11-s + (−5.98 + 0.386i)12-s + (13.3 + 13.3i)13-s + (−2.01 + 9.69i)14-s + (14.1 − 4.94i)15-s + (1.99 + 3.46i)16-s + (−25.3 + 6.77i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (−0.832 + 0.554i)3-s + (0.433 + 0.250i)4-s + (−0.968 − 0.249i)5-s + (0.669 − 0.226i)6-s + (−0.0572 − 0.998i)7-s + (−0.249 − 0.250i)8-s + (0.384 − 0.923i)9-s + (0.615 + 0.347i)10-s + (0.382 + 0.221i)11-s + (−0.498 + 0.0321i)12-s + (1.02 + 1.02i)13-s + (−0.143 + 0.692i)14-s + (0.944 − 0.329i)15-s + (0.124 + 0.216i)16-s + (−1.48 + 0.398i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.688 - 0.725i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.688 - 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.688 - 0.725i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (17, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ 0.688 - 0.725i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.570485 + 0.245054i\)
\(L(\frac12)\)  \(\approx\)  \(0.570485 + 0.245054i\)
\(L(2)\)   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 \( 1 + (1.36 + 0.366i)T \)
3 \( 1 + (2.49 - 1.66i)T \)
5 \( 1 + (4.84 + 1.24i)T \)
7 \( 1 + (0.400 + 6.98i)T \)
good11 \( 1 + (-4.21 - 2.43i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-13.3 - 13.3i)T + 169iT^{2} \)
17 \( 1 + (25.3 - 6.77i)T + (250. - 144.5i)T^{2} \)
19 \( 1 + (-6.44 - 11.1i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (5.04 - 18.8i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 - 41.0T + 841T^{2} \)
31 \( 1 + (-28.4 - 16.4i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-16.1 + 60.2i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 50.2T + 1.68e3T^{2} \)
43 \( 1 + (39.5 - 39.5i)T - 1.84e3iT^{2} \)
47 \( 1 + (0.928 - 3.46i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-55.8 + 14.9i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (-41.0 - 23.7i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (58.7 - 33.9i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-40.3 + 10.7i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 30.7iT - 5.04e3T^{2} \)
73 \( 1 + (-52.4 + 14.0i)T + (4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (26.6 - 15.4i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (43.6 - 43.6i)T - 6.88e3iT^{2} \)
89 \( 1 + (54.3 - 31.3i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-16.7 + 16.7i)T - 9.40e3iT^{2} \)
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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

−11.81733013666863184676936141868, −11.24236069228148592498581768928, −10.49836723494842363909554090079, −9.391476686522111349850750176612, −8.433834350938909611214113307398, −7.14130846358268773091623524319, −6.32825843671588100665886562550, −4.43586132731628184373518175723, −3.80299050558035365148845100949, −1.06903779430786466854238270030, 0.63016130238795948751941342151, 2.70677020035373568716684992201, 4.71296706190501626304595169580, 6.13729621409091076939463829647, 6.80256354149670698842213541651, 8.136842106819828843755199191332, 8.692186787168617583126973483754, 10.25947370813690732704471709064, 11.25006422261559614490545332667, 11.70976181280244025884389632284

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