Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $-0.902 - 0.430i$
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 + (1.06 − 2.80i)3-s + (1.73 + i)4-s + (−0.695 + 4.95i)5-s + (−2.47 + 3.44i)6-s + (−2.48 + 6.54i)7-s + (−1.99 − 2i)8-s + (−6.75 − 5.95i)9-s + (2.76 − 6.50i)10-s + (−16.0 − 9.29i)11-s + (4.64 − 3.80i)12-s + (−11.8 − 11.8i)13-s + (5.79 − 8.02i)14-s + (13.1 + 7.20i)15-s + (1.99 + 3.46i)16-s + (−16.3 + 4.38i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (0.353 − 0.935i)3-s + (0.433 + 0.250i)4-s + (−0.139 + 0.990i)5-s + (−0.412 + 0.574i)6-s + (−0.355 + 0.934i)7-s + (−0.249 − 0.250i)8-s + (−0.750 − 0.661i)9-s + (0.276 − 0.650i)10-s + (−1.46 − 0.844i)11-s + (0.386 − 0.316i)12-s + (−0.912 − 0.912i)13-s + (0.413 − 0.573i)14-s + (0.877 + 0.480i)15-s + (0.124 + 0.216i)16-s + (−0.962 + 0.257i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.902 - 0.430i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.902 - 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.902 - 0.430i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (17, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ -0.902 - 0.430i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.00324519 + 0.0143284i\)
\(L(\frac12)\)  \(\approx\)  \(0.00324519 + 0.0143284i\)
\(L(2)\)   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 \( 1 + (-1.06 + 2.80i)T \)
5 \( 1 + (0.695 - 4.95i)T \)
7 \( 1 + (2.48 - 6.54i)T \)
good11 \( 1 + (16.0 + 9.29i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (11.8 + 11.8i)T + 169iT^{2} \)
17 \( 1 + (16.3 - 4.38i)T + (250. - 144.5i)T^{2} \)
19 \( 1 + (2.06 + 3.58i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (5.44 - 20.3i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 - 49.5T + 841T^{2} \)
31 \( 1 + (2.73 + 1.57i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-8.12 + 30.3i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 26.6T + 1.68e3T^{2} \)
43 \( 1 + (25.3 - 25.3i)T - 1.84e3iT^{2} \)
47 \( 1 + (13.1 - 49.0i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (37.7 - 10.1i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (-46.6 - 26.9i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-34.2 + 19.7i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-47.4 + 12.7i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 81.2iT - 5.04e3T^{2} \)
73 \( 1 + (0.400 - 0.107i)T + (4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (117. - 67.6i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (27.5 - 27.5i)T - 6.88e3iT^{2} \)
89 \( 1 + (20.2 - 11.7i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-50.1 + 50.1i)T - 9.40e3iT^{2} \)
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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

−11.50677866492757936487190192422, −10.66054227280270528207705726141, −9.586215953211861079011465855362, −8.341338415885873784155229431242, −7.75348655974213278805541960960, −6.62284410823537674861105119241, −5.63799805077365559719609400882, −2.98911572447131029983665640815, −2.47866869581004119452225407888, −0.008831058109459105085072742104, 2.41062245139821092429549561550, 4.35130471002536772591656912742, 5.01435873201682753713131952294, 6.83201128896769874623224721661, 7.954178641585308895411197969902, 8.781116327759262951986761318704, 9.931254432559052672648733268248, 10.22473446276261505434357998614, 11.51127844277617431912070918870, 12.70037089854659145000944146518

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