Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.562 - 0.826i$
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 + (2.07 + 1.63i)7-s − 0.999i·8-s + (−5.48 − 3.16i)11-s − 1.05i·13-s + (2.61 + 0.381i)14-s + (−0.5 − 0.866i)16-s + (−2.31 + 4.01i)17-s + (−5.35 + 3.08i)19-s − 6.33·22-s + (−6.10 + 3.52i)23-s + (−0.526 − 0.911i)26-s + (2.45 − 0.978i)28-s − 2.98i·29-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (0.784 + 0.619i)7-s − 0.353i·8-s + (−1.65 − 0.954i)11-s − 0.292i·13-s + (0.699 + 0.101i)14-s + (−0.125 − 0.216i)16-s + (−0.562 + 0.973i)17-s + (−1.22 + 0.708i)19-s − 1.34·22-s + (−1.27 + 0.734i)23-s + (−0.103 − 0.178i)26-s + (0.464 − 0.184i)28-s − 0.553i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.562 - 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.562 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.562 - 0.826i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1151, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.562 - 0.826i)$
$L(1)$  $\approx$  $0.6054841748$
$L(\frac12)$  $\approx$  $0.6054841748$
$L(\frac{3}{2})$   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 \( 1 + (-2.07 - 1.63i)T \)
good11 \( 1 + (5.48 + 3.16i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.05iT - 13T^{2} \)
17 \( 1 + (2.31 - 4.01i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.35 - 3.08i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.10 - 3.52i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.98iT - 29T^{2} \)
31 \( 1 + (-5.46 - 3.15i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.73 - 3.01i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.97T + 41T^{2} \)
43 \( 1 + 2.58T + 43T^{2} \)
47 \( 1 + (-4.07 - 7.06i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (7.68 + 4.43i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.452 - 0.783i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.81 - 5.08i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.91 + 8.51i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 14.4iT - 71T^{2} \)
73 \( 1 + (7.24 + 4.18i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.73 + 4.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 14.6T + 83T^{2} \)
89 \( 1 + (1.91 + 3.32i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 5.87iT - 97T^{2} \)
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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

−8.664928386562692810216949187611, −8.215272893676437396905084804863, −7.69283681773244107516798366880, −6.20622946468898588911063130980, −5.95470706481835108842708877461, −5.05952604744952650350958755140, −4.35735592743326502724830075966, −3.32777158308025850782004348813, −2.44409635615207468253546296563, −1.64253992010601260890739463080, 0.13174755142658798808015238932, 2.04110128191501903162708416795, 2.60095414167363234347383220579, 3.96954202516907647942447429978, 4.80834251777440758160151346534, 4.95392126777732561243732957297, 6.22328712270280912588707575710, 6.93299931051307101298884183041, 7.64576268673768279107967392609, 8.156206596796753396851517671267

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