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  :  $\chi_{525} (157, \cdot )$
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

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

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