Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $-0.897 - 0.440i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.366 + 1.36i)2-s + (0.448 + 1.67i)3-s + (−1.73 − i)4-s + (0.678 + 4.95i)5-s − 2.44·6-s + (5.93 + 3.71i)7-s + (2 − 1.99i)8-s + (−2.59 + 1.50i)9-s + (−7.01 − 0.886i)10-s + (−3.58 + 6.20i)11-s + (0.896 − 3.34i)12-s + (7.28 − 7.28i)13-s + (−7.24 + 6.74i)14-s + (−7.98 + 3.35i)15-s + (1.99 + 3.46i)16-s + (−18.1 + 4.87i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (0.149 + 0.557i)3-s + (−0.433 − 0.250i)4-s + (0.135 + 0.990i)5-s − 0.408·6-s + (0.847 + 0.530i)7-s + (0.250 − 0.249i)8-s + (−0.288 + 0.166i)9-s + (−0.701 − 0.0886i)10-s + (−0.325 + 0.564i)11-s + (0.0747 − 0.278i)12-s + (0.560 − 0.560i)13-s + (−0.517 + 0.481i)14-s + (−0.532 + 0.223i)15-s + (0.124 + 0.216i)16-s + (−1.06 + 0.286i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.897 - 0.440i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.897 - 0.440i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.897 - 0.440i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (67, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ -0.897 - 0.440i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.309296 + 1.33124i\)
\(L(\frac12)\)  \(\approx\)  \(0.309296 + 1.33124i\)
\(L(2)\)   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 \( 1 + (0.366 - 1.36i)T \)
3 \( 1 + (-0.448 - 1.67i)T \)
5 \( 1 + (-0.678 - 4.95i)T \)
7 \( 1 + (-5.93 - 3.71i)T \)
good11 \( 1 + (3.58 - 6.20i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-7.28 + 7.28i)T - 169iT^{2} \)
17 \( 1 + (18.1 - 4.87i)T + (250. - 144.5i)T^{2} \)
19 \( 1 + (7.65 - 4.41i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-0.401 - 0.107i)T + (458. + 264.5i)T^{2} \)
29 \( 1 - 27.7iT - 841T^{2} \)
31 \( 1 + (-12.5 + 21.6i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-13.8 + 51.5i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 46.7T + 1.68e3T^{2} \)
43 \( 1 + (37.3 - 37.3i)T - 1.84e3iT^{2} \)
47 \( 1 + (-8.32 + 31.0i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-18.4 - 68.8i)T + (-2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (-35.2 - 20.3i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-3.30 - 5.72i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-128. + 34.4i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 129.T + 5.04e3T^{2} \)
73 \( 1 + (-2.10 - 7.85i)T + (-4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (-59.9 + 34.6i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-57.0 + 57.0i)T - 6.88e3iT^{2} \)
89 \( 1 + (114. - 65.8i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (46.0 + 46.0i)T + 9.40e3iT^{2} \)
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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

−12.67623407945049500251645221635, −11.18392149639196311061838100799, −10.65693947356668114974666196152, −9.535793053117276117298906244135, −8.497396111448581029377415364005, −7.60235802015414069096522306133, −6.37440097436623355816701090203, −5.32880298919208214826090968027, −4.05011156961553600062886998711, −2.34111054375010135518648298756, 0.833836080486318294907237642833, 2.15764996495389873856212065434, 4.04932976888020258338414522213, 5.11105170244594485522627691913, 6.64432094642354228208961724467, 8.144426297254965293928169698941, 8.572687977583754542815058579580, 9.719141348772686390283992485417, 11.04990137218754295309357707246, 11.59873591552447112665432000227

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