Properties

Degree 2
Conductor $ 3 \cdot 5^{2} \cdot 7 $
Sign $-0.691 + 0.722i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.448 − 1.67i)3-s + (−1.73 + i)4-s + (1.48 − 2.19i)7-s + (−2.59 − 1.50i)9-s + (0.896 + 3.34i)12-s + (−1.22 − 1.22i)13-s + (1.99 − 3.46i)16-s + (−6.06 − 3.5i)19-s + (−2.99 − 3.46i)21-s + (−3.67 + 3.67i)27-s + (−0.378 + 5.27i)28-s + (−3.5 − 6.06i)31-s + 6·36-s + (−3.13 − 11.7i)37-s + (−2.59 + 1.5i)39-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)3-s + (−0.866 + 0.5i)4-s + (0.560 − 0.827i)7-s + (−0.866 − 0.5i)9-s + (0.258 + 0.965i)12-s + (−0.339 − 0.339i)13-s + (0.499 − 0.866i)16-s + (−1.39 − 0.802i)19-s + (−0.654 − 0.755i)21-s + (−0.707 + 0.707i)27-s + (−0.0716 + 0.997i)28-s + (−0.628 − 1.08i)31-s + 36-s + (−0.515 − 1.92i)37-s + (−0.416 + 0.240i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.691 + 0.722i$
motivic weight  =  \(1\)
character  :  $\chi_{525} (443, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 525,\ (\ :1/2),\ -0.691 + 0.722i)$
$L(1)$  $\approx$  $0.369305 - 0.864945i$
$L(\frac12)$  $\approx$  $0.369305 - 0.864945i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + (-0.448 + 1.67i)T \)
5 \( 1 \)
7 \( 1 + (-1.48 + 2.19i)T \)
good2 \( 1 + (1.73 - i)T^{2} \)
11 \( 1 + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.22 + 1.22i)T + 13iT^{2} \)
17 \( 1 + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (6.06 + 3.5i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (3.5 + 6.06i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.13 + 11.7i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (-8.57 - 8.57i)T + 43iT^{2} \)
47 \( 1 + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7 - 12.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-11.7 - 3.13i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-4.03 + 15.0i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-11.2 - 6.5i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.79 + 9.79i)T - 97iT^{2} \)
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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

−10.65016930468359217156854084225, −9.376392964740005742172797270680, −8.642759739138032115581087129630, −7.74335955796890403306695717712, −7.24231427510446277359406856045, −5.95903591807952462892729985923, −4.69901619515863476200878258026, −3.73048005735963774232862522447, −2.30473674086684441801836637953, −0.53980730765306401063073339251, 2.04362199664922667578660290461, 3.62317789869252174946665462805, 4.67133038913049182090238152030, 5.30701936591689253515148777312, 6.33032933234732601852404954573, 8.060133350213531141283826826409, 8.701241168958629354226342021688, 9.344868405127494884955120469196, 10.26838486904554545806768916937, 10.88825857109180073812063615686

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