Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 7 $
Sign $0.131 + 0.991i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + (−0.554 + 1.64i)3-s + 4-s + (−0.5 + 0.866i)5-s + (0.554 − 1.64i)6-s + (2.32 − 1.26i)7-s − 8-s + (−2.38 − 1.82i)9-s + (0.5 − 0.866i)10-s + (−1.25 − 2.17i)11-s + (−0.554 + 1.64i)12-s + (−2.97 − 5.15i)13-s + (−2.32 + 1.26i)14-s + (−1.14 − 1.30i)15-s + 16-s + (−2.65 + 4.60i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.320 + 0.947i)3-s + 0.5·4-s + (−0.223 + 0.387i)5-s + (0.226 − 0.669i)6-s + (0.878 − 0.478i)7-s − 0.353·8-s + (−0.794 − 0.606i)9-s + (0.158 − 0.273i)10-s + (−0.378 − 0.655i)11-s + (−0.160 + 0.473i)12-s + (−0.826 − 1.43i)13-s + (−0.621 + 0.338i)14-s + (−0.295 − 0.335i)15-s + 0.250·16-s + (−0.644 + 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.131 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.131 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.131 + 0.991i$
motivic weight  =  \(1\)
character  :  $\chi_{630} (151, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 630,\ (\ :1/2),\ 0.131 + 0.991i)$
$L(1)$  $\approx$  $0.354982 - 0.310988i$
$L(\frac12)$  $\approx$  $0.354982 - 0.310988i$
$L(\frac{3}{2})$   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 \( 1 + (0.554 - 1.64i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-2.32 + 1.26i)T \)
good11 \( 1 + (1.25 + 2.17i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.97 + 5.15i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.65 - 4.60i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.97 + 5.15i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.678 - 1.17i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.10 + 5.37i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.16T + 31T^{2} \)
37 \( 1 + (-2.58 - 4.47i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.43 + 7.67i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.977 + 1.69i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.04T + 47T^{2} \)
53 \( 1 + (0.179 - 0.311i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 2.86T + 59T^{2} \)
61 \( 1 - 3.82T + 61T^{2} \)
67 \( 1 - 12.9T + 67T^{2} \)
71 \( 1 - 3.52T + 71T^{2} \)
73 \( 1 + (-5.60 + 9.70i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 14.7T + 79T^{2} \)
83 \( 1 + (2.54 - 4.39i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.367 + 0.636i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.19 - 10.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.61451270559045531086497646184, −9.718936468410748202562684327450, −8.533360095777450837298754540223, −8.065278615981632817266079945292, −6.94211461734179015662377094381, −5.80586757433346989844974360477, −4.86679431164296218681147637882, −3.72716586831903467338445632521, −2.49131979764861365838878050463, −0.32459048819994495453110198454, 1.63142191599702669803506430823, 2.40648433876455633147101083518, 4.51506148760880498109098087614, 5.37735521535239739603748940748, 6.65386449793851428194079089464, 7.32224291695607909719173499190, 8.157860640696354402713354179662, 8.888060091222764247543004367444, 9.795057244480259852692239375354, 11.02925781874498349783332504630

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