Properties

Degree 2
Conductor $ 3 \cdot 5^{2} \cdot 7 $
Sign $0.925 + 0.379i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.965 + 0.258i)3-s + (1.73 − i)4-s + (2.63 + 0.189i)7-s + (0.866 + 0.499i)9-s + (−3 − 5.19i)11-s + (1.93 − 0.517i)12-s + (1.41 − 1.41i)13-s + (1.99 − 3.46i)16-s + (−1.55 + 5.79i)17-s + (−1.73 + 3i)19-s + (2.50 + 0.866i)21-s + (0.707 + 0.707i)27-s + (4.76 − 2.31i)28-s + (−4.5 + 2.59i)31-s + (−1.55 − 5.79i)33-s + ⋯
L(s)  = 1  + (0.557 + 0.149i)3-s + (0.866 − 0.5i)4-s + (0.997 + 0.0716i)7-s + (0.288 + 0.166i)9-s + (−0.904 − 1.56i)11-s + (0.557 − 0.149i)12-s + (0.392 − 0.392i)13-s + (0.499 − 0.866i)16-s + (−0.376 + 1.40i)17-s + (−0.397 + 0.688i)19-s + (0.545 + 0.188i)21-s + (0.136 + 0.136i)27-s + (0.899 − 0.436i)28-s + (−0.808 + 0.466i)31-s + (−0.270 − 1.00i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.379i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.925 + 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.925 + 0.379i$
motivic weight  =  \(1\)
character  :  $\chi_{525} (418, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 525,\ (\ :1/2),\ 0.925 + 0.379i)$
$L(1)$  $\approx$  $2.15312 - 0.424726i$
$L(\frac12)$  $\approx$  $2.15312 - 0.424726i$
$L(\frac{3}{2})$   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 \( 1 + (-0.965 - 0.258i)T \)
5 \( 1 \)
7 \( 1 + (-2.63 - 0.189i)T \)
good2 \( 1 + (-1.73 + i)T^{2} \)
11 \( 1 + (3 + 5.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.41 + 1.41i)T - 13iT^{2} \)
17 \( 1 + (1.55 - 5.79i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.73 - 3i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (4.5 - 2.59i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.24 - 8.36i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 10.3iT - 41T^{2} \)
43 \( 1 + (3.67 + 3.67i)T + 43iT^{2} \)
47 \( 1 + (-5.79 + 1.55i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.68 - 10.0i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-5.19 - 9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.5 - 4.33i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.34 + 0.896i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (0.965 + 0.258i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (4.33 + 2.5i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.24 - 4.24i)T - 83iT^{2} \)
89 \( 1 + (5.19 - 9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (9.19 + 9.19i)T + 97iT^{2} \)
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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

−10.61988233080584298314447886426, −10.38072116973796158938560158694, −8.677638170458938821893565057164, −8.300928967097722086012210589286, −7.32930220466466157147377042008, −5.99885065638355380682735979796, −5.43215520181488198586010584735, −3.89747800404006669862979237514, −2.69957126160765208841230825974, −1.47925147411753385254175920294, 1.90850787529175323334230912388, 2.65253997425464728676013197441, 4.19489773233159367548112308722, 5.13034921189639060721123096555, 6.71245122667750072721757476048, 7.40827966507634255495875935229, 7.992494638560715196097247573528, 9.062893215142945898159699605499, 10.03480983960659418656922171234, 11.15624346787031546070107947014

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