Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.153 + 0.988i$
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 + (0.499 − 0.866i)4-s + (−0.397 − 2.61i)7-s + 0.999i·8-s + (0.429 + 0.248i)11-s − 2.74i·13-s + (1.65 + 2.06i)14-s + (−0.5 − 0.866i)16-s + (1.82 − 3.16i)17-s + (3.12 − 1.80i)19-s − 0.496·22-s + (5.56 − 3.21i)23-s + (1.37 + 2.37i)26-s + (−2.46 − 0.963i)28-s + 8.87i·29-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.150 − 0.988i)7-s + 0.353i·8-s + (0.129 + 0.0748i)11-s − 0.761i·13-s + (0.441 + 0.552i)14-s + (−0.125 − 0.216i)16-s + (0.443 − 0.768i)17-s + (0.716 − 0.413i)19-s − 0.105·22-s + (1.16 − 0.669i)23-s + (0.269 + 0.466i)26-s + (−0.465 − 0.182i)28-s + 1.64i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.153 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.153 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.153 + 0.988i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1151, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.153 + 0.988i)$
$L(1)$  $\approx$  $1.049805567$
$L(\frac12)$  $\approx$  $1.049805567$
$L(\frac{3}{2})$   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 \( 1 \)
5 \( 1 \)
7 \( 1 + (0.397 + 2.61i)T \)
good11 \( 1 + (-0.429 - 0.248i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.74iT - 13T^{2} \)
17 \( 1 + (-1.82 + 3.16i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.12 + 1.80i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.56 + 3.21i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 8.87iT - 29T^{2} \)
31 \( 1 + (6.90 + 3.98i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.14 - 1.98i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.22T + 41T^{2} \)
43 \( 1 + 2.22T + 43T^{2} \)
47 \( 1 + (3.27 + 5.66i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.72 - 3.88i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.05 - 5.28i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.24 + 1.87i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.08 + 7.08i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.3iT - 71T^{2} \)
73 \( 1 + (11.3 + 6.53i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.44 - 7.69i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.79T + 83T^{2} \)
89 \( 1 + (-0.743 - 1.28i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 9.05iT - 97T^{2} \)
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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

−8.491226925311972093819795256149, −7.51847485762069403646399319279, −7.21108098876823263716900927382, −6.47111209936796724519798958622, −5.36481578836441480316465087228, −4.85206976207797317623988209586, −3.59751702900812238358059073180, −2.85164054790960044913961527319, −1.38525245532892692076482261284, −0.43925873302295285400800697193, 1.29690872683334398231679634402, 2.20603309880635324115088330045, 3.20075313425383336246078700646, 3.98012893484767194450886882437, 5.18781963114596749589894526527, 5.87578143833326442264491092416, 6.72891473886944013231475569006, 7.52178035125793698749679069075, 8.309343872894678586497336693489, 8.955262268684977663823648525506

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