Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $0.176 + 0.984i$
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 + (1.67 + 0.448i)3-s + (1.73 − i)4-s + (−4.60 + 1.93i)5-s − 2.44·6-s + (−1.45 − 6.84i)7-s + (−1.99 + 2i)8-s + (2.59 + 1.50i)9-s + (5.58 − 4.33i)10-s + (−6.59 − 11.4i)11-s + (3.34 − 0.896i)12-s + (10.7 − 10.7i)13-s + (4.49 + 8.81i)14-s + (−8.58 + 1.17i)15-s + (1.99 − 3.46i)16-s + (2.10 − 7.85i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.557 + 0.149i)3-s + (0.433 − 0.250i)4-s + (−0.921 + 0.387i)5-s − 0.408·6-s + (−0.208 − 0.978i)7-s + (−0.249 + 0.250i)8-s + (0.288 + 0.166i)9-s + (0.558 − 0.433i)10-s + (−0.599 − 1.03i)11-s + (0.278 − 0.0747i)12-s + (0.828 − 0.828i)13-s + (0.321 + 0.629i)14-s + (−0.572 + 0.0783i)15-s + (0.124 − 0.216i)16-s + (0.123 − 0.462i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.176 + 0.984i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.176 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.176 + 0.984i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (37, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ 0.176 + 0.984i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.657750 - 0.550334i\)
\(L(\frac12)\)  \(\approx\)  \(0.657750 - 0.550334i\)
\(L(2)\)   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 \( 1 + (-1.67 - 0.448i)T \)
5 \( 1 + (4.60 - 1.93i)T \)
7 \( 1 + (1.45 + 6.84i)T \)
good11 \( 1 + (6.59 + 11.4i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (-10.7 + 10.7i)T - 169iT^{2} \)
17 \( 1 + (-2.10 + 7.85i)T + (-250. - 144.5i)T^{2} \)
19 \( 1 + (8.35 + 4.82i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-6.06 - 22.6i)T + (-458. + 264.5i)T^{2} \)
29 \( 1 + 41.4iT - 841T^{2} \)
31 \( 1 + (18.0 + 31.3i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (4.68 - 1.25i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 7.31T + 1.68e3T^{2} \)
43 \( 1 + (-48.7 + 48.7i)T - 1.84e3iT^{2} \)
47 \( 1 + (51.9 - 13.9i)T + (1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-45.4 - 12.1i)T + (2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (1.15 - 0.665i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (29.7 - 51.6i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (31.5 - 117. i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 82.6T + 5.04e3T^{2} \)
73 \( 1 + (-49.5 - 13.2i)T + (4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (26.3 + 15.2i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (32.2 - 32.2i)T - 6.88e3iT^{2} \)
89 \( 1 + (19.1 + 11.0i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-128. - 128. i)T + 9.40e3iT^{2} \)
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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

−11.51806554040516912638643227767, −10.84187243351717152637040050169, −10.04390915945853194788541992304, −8.752802162269406758956037438901, −7.889328269579596388928019273756, −7.25988889014493072934236871400, −5.85006627510998448075193414763, −4.00884894848748822942175222979, −2.97782631652241234769473912829, −0.56555413309403281794656046532, 1.80601220009932819551524143855, 3.31323746787223876093205996990, 4.74721733790357576889339585102, 6.49208474966188194658468548298, 7.57653812571670699430121762375, 8.615083880893734948173293547277, 9.038058432470961190637981031964, 10.35141002582488094900518321705, 11.37859196339881092051484823561, 12.56492289820546539800524586310

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