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  :  $\chi_{525} (107, \cdot )$
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

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

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