Properties

Degree 2
Conductor $ 3 \cdot 5^{2} \cdot 7 $
Sign $0.899 + 0.436i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.800 + 0.800i)2-s + (1.09 − 1.34i)3-s + 0.718i·4-s + (0.199 + 1.95i)6-s + (0.707 + 0.707i)7-s + (−2.17 − 2.17i)8-s + (−0.606 − 2.93i)9-s − 5.20i·11-s + (0.964 + 0.785i)12-s + (3.24 − 3.24i)13-s − 1.13·14-s + 2.04·16-s + (−0.844 + 0.844i)17-s + (2.83 + 1.86i)18-s − 1.32i·19-s + ⋯
L(s)  = 1  + (−0.566 + 0.566i)2-s + (0.631 − 0.775i)3-s + 0.359i·4-s + (0.0813 + 0.796i)6-s + (0.267 + 0.267i)7-s + (−0.769 − 0.769i)8-s + (−0.202 − 0.979i)9-s − 1.56i·11-s + (0.278 + 0.226i)12-s + (0.900 − 0.900i)13-s − 0.302·14-s + 0.511·16-s + (−0.204 + 0.204i)17-s + (0.668 + 0.439i)18-s − 0.302i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 + 0.436i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.899 + 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.899 + 0.436i$
motivic weight  =  \(1\)
character  :  $\chi_{525} (407, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 525,\ (\ :1/2),\ 0.899 + 0.436i)\)
\(L(1)\)  \(\approx\)  \(1.26385 - 0.290459i\)
\(L(\frac12)\)  \(\approx\)  \(1.26385 - 0.290459i\)
\(L(\frac{3}{2})\)   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 \( 1 + (-1.09 + 1.34i)T \)
5 \( 1 \)
7 \( 1 + (-0.707 - 0.707i)T \)
good2 \( 1 + (0.800 - 0.800i)T - 2iT^{2} \)
11 \( 1 + 5.20iT - 11T^{2} \)
13 \( 1 + (-3.24 + 3.24i)T - 13iT^{2} \)
17 \( 1 + (0.844 - 0.844i)T - 17iT^{2} \)
19 \( 1 + 1.32iT - 19T^{2} \)
23 \( 1 + (-5.62 - 5.62i)T + 23iT^{2} \)
29 \( 1 - 4.38T + 29T^{2} \)
31 \( 1 + 1.70T + 31T^{2} \)
37 \( 1 + (-1.71 - 1.71i)T + 37iT^{2} \)
41 \( 1 + 1.82iT - 41T^{2} \)
43 \( 1 + (-0.281 + 0.281i)T - 43iT^{2} \)
47 \( 1 + (-3.39 + 3.39i)T - 47iT^{2} \)
53 \( 1 + (3.51 + 3.51i)T + 53iT^{2} \)
59 \( 1 - 1.81T + 59T^{2} \)
61 \( 1 + 2.47T + 61T^{2} \)
67 \( 1 + (7.92 + 7.92i)T + 67iT^{2} \)
71 \( 1 - 9.06iT - 71T^{2} \)
73 \( 1 + (-1.33 + 1.33i)T - 73iT^{2} \)
79 \( 1 - 11.5iT - 79T^{2} \)
83 \( 1 + (-5.46 - 5.46i)T + 83iT^{2} \)
89 \( 1 + 9.43T + 89T^{2} \)
97 \( 1 + (-3.06 - 3.06i)T + 97iT^{2} \)
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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

−10.89025197209251288550495577149, −9.443334427863854462501532946691, −8.601561523599200721388316551314, −8.301613850374325766269090713017, −7.37988522503672955487980160619, −6.42456666141061126999968015815, −5.60568820735737335806778923518, −3.60961107500506950178092399988, −2.91795012986973501182823694670, −0.943510245689465515549878720812, 1.62225604244533142905325040563, 2.72370793584305510865383277456, 4.25326077523280888306620487289, 4.94445955626716885560057730557, 6.38570557456598227280164359903, 7.54634404613277684762606605623, 8.727872715137183009570752560243, 9.192317827132437010276010903008, 10.08125053904457994045928843555, 10.67882513251379934651353847276

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