Properties

Degree 2
Conductor $ 3 \cdot 5^{2} \cdot 7 $
Sign $0.955 - 0.295i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.67 + 0.448i)3-s + (1.73 + i)4-s + (2.19 − 1.48i)7-s + (2.59 − 1.50i)9-s + (−3.34 − 0.896i)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 + (5.27 − 0.378i)28-s + (−3.5 + 6.06i)31-s + 6·36-s + (11.7 + 3.13i)37-s + (2.59 + 1.5i)39-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)3-s + (0.866 + 0.5i)4-s + (0.827 − 0.560i)7-s + (0.866 − 0.5i)9-s + (−0.965 − 0.258i)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 + (0.997 − 0.0716i)28-s + (−0.628 + 1.08i)31-s + 36-s + (1.92 + 0.515i)37-s + (0.416 + 0.240i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.295i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 - 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.955 - 0.295i$
motivic weight  =  \(1\)
character  :  $\chi_{525} (368, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 525,\ (\ :1/2),\ 0.955 - 0.295i)$
$L(1)$  $\approx$  $1.43753 + 0.217316i$
$L(\frac12)$  $\approx$  $1.43753 + 0.217316i$
$L(\frac{3}{2})$   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 \( 1 + (1.67 - 0.448i)T \)
5 \( 1 \)
7 \( 1 + (-2.19 + 1.48i)T \)
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 \( 1 + (-11.7 - 3.13i)T + (32.0 + 18.5i)T^{2} \)
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 \( 1 + (3.13 + 11.7i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (15.0 - 4.03i)T + (63.2 - 36.5i)T^{2} \)
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

−11.13654951646965371887672011508, −10.30407583087062455189413369439, −9.295348708221828095170865104290, −7.84246470978865197057808464576, −7.32893633882001626295505901098, −6.35282046371830974544277676975, −5.27195692839910834608986886361, −4.33630968691001933827462736909, −3.00079923207677903408503778244, −1.27633608841095443761882035803, 1.28341604350684798804014356270, 2.44345824704567577083851106120, 4.35748836276370702660484647474, 5.61136797643005910361167270684, 5.88218410745803144290031852142, 7.27604684773957351383588095886, 7.69974674506343887471577399191, 9.239329450870038638276424057868, 10.16438767063170635877875257108, 10.99163922961575457156627610041

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