Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 7 $
Sign $0.743 + 0.668i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)2-s + (1.16 + 1.28i)3-s + (0.499 + 0.866i)4-s − 5-s + (−0.363 − 1.69i)6-s + (−1.57 − 2.12i)7-s − 0.999i·8-s + (−0.300 + 2.98i)9-s + (0.866 + 0.5i)10-s − 5.66i·11-s + (−0.531 + 1.64i)12-s + (2.43 + 1.40i)13-s + (0.300 + 2.62i)14-s + (−1.16 − 1.28i)15-s + (−0.5 + 0.866i)16-s + (3.51 − 6.08i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.670 + 0.741i)3-s + (0.249 + 0.433i)4-s − 0.447·5-s + (−0.148 − 0.691i)6-s + (−0.595 − 0.803i)7-s − 0.353i·8-s + (−0.100 + 0.994i)9-s + (0.273 + 0.158i)10-s − 1.70i·11-s + (−0.153 + 0.475i)12-s + (0.675 + 0.389i)13-s + (0.0802 + 0.702i)14-s + (−0.299 − 0.331i)15-s + (−0.125 + 0.216i)16-s + (0.851 − 1.47i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.743 + 0.668i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.743 + 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.743 + 0.668i$
motivic weight  =  \(1\)
character  :  $\chi_{630} (311, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 630,\ (\ :1/2),\ 0.743 + 0.668i)\)
\(L(1)\)  \(\approx\)  \(1.12073 - 0.429891i\)
\(L(\frac12)\)  \(\approx\)  \(1.12073 - 0.429891i\)
\(L(\frac{3}{2})\)   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 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (-1.16 - 1.28i)T \)
5 \( 1 + T \)
7 \( 1 + (1.57 + 2.12i)T \)
good11 \( 1 + 5.66iT - 11T^{2} \)
13 \( 1 + (-2.43 - 1.40i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.51 + 6.08i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.76 + 2.17i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 4.07iT - 23T^{2} \)
29 \( 1 + (-5.28 + 3.05i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.319 + 0.184i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.783 - 1.35i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.39 + 2.41i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.18 - 5.51i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.59 - 4.49i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.02 + 1.74i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.32 + 9.22i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-12.6 - 7.32i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.36 + 2.36i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.06iT - 71T^{2} \)
73 \( 1 + (14.3 + 8.29i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.37 + 7.57i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.34 + 14.4i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.332 + 0.575i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.68 + 0.971i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.38162896763129200903481273892, −9.557985663384092680984853233070, −8.954890372458604895601602279793, −7.988506338805154033717915213479, −7.33990186950095305224494380058, −6.02910099153138745460115116424, −4.65247364642649624939577683994, −3.37479593062257064099984001055, −3.09507656580674923487906688515, −0.840684990308132201785092990972, 1.42062411935293469990971564488, 2.69553675201294553639723591588, 3.92036572661949507405223646244, 5.53860649020315935486030198094, 6.49522295032316614631222914020, 7.27044813306897666805785246889, 8.163987677745565236541973337188, 8.700556739698713439688346191971, 9.750080334056334594631445935780, 10.31889089992832998845130966216

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