Properties

Degree 2
Conductor $ 3 \cdot 5^{2} \cdot 7 $
Sign $0.868 - 0.496i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.79 + 1.79i)2-s + (−0.491 − 1.66i)3-s − 4.47i·4-s + (3.87 + 2.10i)6-s + (0.707 + 0.707i)7-s + (4.45 + 4.45i)8-s + (−2.51 + 1.63i)9-s + 1.56i·11-s + (−7.43 + 2.19i)12-s + (−2.21 + 2.21i)13-s − 2.54·14-s − 7.09·16-s + (3.60 − 3.60i)17-s + (1.59 − 7.46i)18-s − 1.68i·19-s + ⋯
L(s)  = 1  + (−1.27 + 1.27i)2-s + (−0.283 − 0.958i)3-s − 2.23i·4-s + (1.58 + 0.859i)6-s + (0.267 + 0.267i)7-s + (1.57 + 1.57i)8-s + (−0.839 + 0.543i)9-s + 0.472i·11-s + (−2.14 + 0.634i)12-s + (−0.615 + 0.615i)13-s − 0.680·14-s − 1.77·16-s + (0.874 − 0.874i)17-s + (0.375 − 1.75i)18-s − 0.385i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.868 - 0.496i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.868 - 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.868 - 0.496i$
motivic weight  =  \(1\)
character  :  $\chi_{525} (407, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 525,\ (\ :1/2),\ 0.868 - 0.496i)\)
\(L(1)\)  \(\approx\)  \(0.619902 + 0.164697i\)
\(L(\frac12)\)  \(\approx\)  \(0.619902 + 0.164697i\)
\(L(\frac{3}{2})\)   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 \( 1 + (0.491 + 1.66i)T \)
5 \( 1 \)
7 \( 1 + (-0.707 - 0.707i)T \)
good2 \( 1 + (1.79 - 1.79i)T - 2iT^{2} \)
11 \( 1 - 1.56iT - 11T^{2} \)
13 \( 1 + (2.21 - 2.21i)T - 13iT^{2} \)
17 \( 1 + (-3.60 + 3.60i)T - 17iT^{2} \)
19 \( 1 + 1.68iT - 19T^{2} \)
23 \( 1 + (-0.995 - 0.995i)T + 23iT^{2} \)
29 \( 1 - 8.91T + 29T^{2} \)
31 \( 1 - 2.74T + 31T^{2} \)
37 \( 1 + (0.440 + 0.440i)T + 37iT^{2} \)
41 \( 1 + 6.44iT - 41T^{2} \)
43 \( 1 + (-5.47 + 5.47i)T - 43iT^{2} \)
47 \( 1 + (-3.69 + 3.69i)T - 47iT^{2} \)
53 \( 1 + (-2.83 - 2.83i)T + 53iT^{2} \)
59 \( 1 + 5.54T + 59T^{2} \)
61 \( 1 - 7.40T + 61T^{2} \)
67 \( 1 + (-3.75 - 3.75i)T + 67iT^{2} \)
71 \( 1 - 3.61iT - 71T^{2} \)
73 \( 1 + (-5.89 + 5.89i)T - 73iT^{2} \)
79 \( 1 - 17.0iT - 79T^{2} \)
83 \( 1 + (-3.21 - 3.21i)T + 83iT^{2} \)
89 \( 1 - 9.40T + 89T^{2} \)
97 \( 1 + (4.39 + 4.39i)T + 97iT^{2} \)
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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

−10.67024220694512061716712619674, −9.753523059154297700782126481308, −8.901507389186640192711916555151, −8.109027907110932073363327527419, −7.22651504580154973562547676906, −6.83232864300312333589839920272, −5.70439818221018128223873778460, −4.91760912830430285206552319687, −2.35077446527450845681041609929, −0.874964456220734838023374145107, 0.928592873889060889421495769307, 2.73248835275547646897003079619, 3.62610738137239562616302176282, 4.79476583010762708293791330343, 6.17657499580964066114691456398, 7.79838600591418476169755745578, 8.367091505589450882946507339518, 9.308170609823339898474246235257, 10.22526862982835275717895662836, 10.43813416878552193044249994225

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