Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 7 $
Sign $0.0218 - 0.999i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + (0.478 − 1.66i)3-s + 4-s + (−0.5 + 0.866i)5-s + (−0.478 + 1.66i)6-s + (−2.56 + 0.658i)7-s − 8-s + (−2.54 − 1.59i)9-s + (0.5 − 0.866i)10-s + (1.11 + 1.93i)11-s + (0.478 − 1.66i)12-s + (0.263 + 0.456i)13-s + (2.56 − 0.658i)14-s + (1.20 + 1.24i)15-s + 16-s + (−2.56 + 4.44i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.275 − 0.961i)3-s + 0.5·4-s + (−0.223 + 0.387i)5-s + (−0.195 + 0.679i)6-s + (−0.968 + 0.249i)7-s − 0.353·8-s + (−0.847 − 0.530i)9-s + (0.158 − 0.273i)10-s + (0.336 + 0.583i)11-s + (0.137 − 0.480i)12-s + (0.0730 + 0.126i)13-s + (0.684 − 0.176i)14-s + (0.310 + 0.321i)15-s + 0.250·16-s + (−0.621 + 1.07i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0218 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0218 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.0218 - 0.999i$
motivic weight  =  \(1\)
character  :  $\chi_{630} (151, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 630,\ (\ :1/2),\ 0.0218 - 0.999i)$
$L(1)$  $\approx$  $0.370670 + 0.362667i$
$L(\frac12)$  $\approx$  $0.370670 + 0.362667i$
$L(\frac{3}{2})$   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 \( 1 + T \)
3 \( 1 + (-0.478 + 1.66i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (2.56 - 0.658i)T \)
good11 \( 1 + (-1.11 - 1.93i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.263 - 0.456i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.56 - 4.44i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.263 - 0.456i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.82 - 6.62i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.08 + 1.87i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 0.275T + 31T^{2} \)
37 \( 1 + (1.07 + 1.86i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.55 - 9.61i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.51 + 2.62i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 0.971T + 47T^{2} \)
53 \( 1 + (5.80 - 10.0i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 9.37T + 59T^{2} \)
61 \( 1 - 2.10T + 61T^{2} \)
67 \( 1 - 2.64T + 67T^{2} \)
71 \( 1 + 0.00533T + 71T^{2} \)
73 \( 1 + (2.19 - 3.79i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 12.0T + 79T^{2} \)
83 \( 1 + (-7.55 + 13.0i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.23 - 12.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.85 - 4.95i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.77987066620113393224912931838, −9.717282494770504811711664234047, −9.109156871323277001064275083528, −8.082616207629900243543792321512, −7.37463926654908535406884513184, −6.47004107905224605031930313494, −5.93027872855666779256201335620, −3.92169751214699775071029945134, −2.79528406747245815343377405370, −1.62850325287363171828606148187, 0.33296095607326357615030538926, 2.59027166148629034227906074434, 3.62161672504840049082330596732, 4.68139159897792839240394154043, 5.92924854098899143601392530684, 6.87617565819794384168353003856, 8.049449628651490276902939693977, 8.879826809179602543229425770443, 9.407852298088939560239495485477, 10.25828439834151323991243008879

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