Properties

Degree 2
Conductor $ 3 \cdot 5^{2} \cdot 7 $
Sign $-0.991 + 0.129i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.347 + 0.347i)2-s + (−0.176 + 1.72i)3-s + 1.75i·4-s + (−0.536 − 0.659i)6-s + (0.707 + 0.707i)7-s + (−1.30 − 1.30i)8-s + (−2.93 − 0.607i)9-s + 2.67i·11-s + (−3.03 − 0.310i)12-s + (−2.14 + 2.14i)13-s − 0.490·14-s − 2.61·16-s + (3.26 − 3.26i)17-s + (1.23 − 0.808i)18-s + 5.24i·19-s + ⋯
L(s)  = 1  + (−0.245 + 0.245i)2-s + (−0.101 + 0.994i)3-s + 0.879i·4-s + (−0.219 − 0.269i)6-s + (0.267 + 0.267i)7-s + (−0.461 − 0.461i)8-s + (−0.979 − 0.202i)9-s + 0.805i·11-s + (−0.874 − 0.0895i)12-s + (−0.596 + 0.596i)13-s − 0.131·14-s − 0.653·16-s + (0.792 − 0.792i)17-s + (0.290 − 0.190i)18-s + 1.20i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.129i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.991 + 0.129i$
motivic weight  =  \(1\)
character  :  $\chi_{525} (407, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 525,\ (\ :1/2),\ -0.991 + 0.129i)\)
\(L(1)\)  \(\approx\)  \(0.0578050 - 0.889395i\)
\(L(\frac12)\)  \(\approx\)  \(0.0578050 - 0.889395i\)
\(L(\frac{3}{2})\)   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 \( 1 + (0.176 - 1.72i)T \)
5 \( 1 \)
7 \( 1 + (-0.707 - 0.707i)T \)
good2 \( 1 + (0.347 - 0.347i)T - 2iT^{2} \)
11 \( 1 - 2.67iT - 11T^{2} \)
13 \( 1 + (2.14 - 2.14i)T - 13iT^{2} \)
17 \( 1 + (-3.26 + 3.26i)T - 17iT^{2} \)
19 \( 1 - 5.24iT - 19T^{2} \)
23 \( 1 + (2.54 + 2.54i)T + 23iT^{2} \)
29 \( 1 - 2.86T + 29T^{2} \)
31 \( 1 + 5.28T + 31T^{2} \)
37 \( 1 + (-2.14 - 2.14i)T + 37iT^{2} \)
41 \( 1 + 11.5iT - 41T^{2} \)
43 \( 1 + (0.759 - 0.759i)T - 43iT^{2} \)
47 \( 1 + (7.66 - 7.66i)T - 47iT^{2} \)
53 \( 1 + (-4.43 - 4.43i)T + 53iT^{2} \)
59 \( 1 + 0.159T + 59T^{2} \)
61 \( 1 - 4.72T + 61T^{2} \)
67 \( 1 + (-5.41 - 5.41i)T + 67iT^{2} \)
71 \( 1 - 13.5iT - 71T^{2} \)
73 \( 1 + (4.16 - 4.16i)T - 73iT^{2} \)
79 \( 1 - 3.89iT - 79T^{2} \)
83 \( 1 + (-4.03 - 4.03i)T + 83iT^{2} \)
89 \( 1 + 3.95T + 89T^{2} \)
97 \( 1 + (-1.86 - 1.86i)T + 97iT^{2} \)
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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

−11.41406587883733643383800323418, −10.16133333696140784887803870648, −9.585383888395394402591184826396, −8.704384337275815957014556958035, −7.84681897800902392155409063331, −6.93722905080648129351772093447, −5.66146056648622038419619847839, −4.59110211311950521361434362964, −3.71919038909801863179523266209, −2.46329043257709954332259843282, 0.56416466967405150590560353420, 1.83775242008827093022133714387, 3.15279451079465696575170468228, 5.00713020330344285800526170802, 5.80924863871799469605682983046, 6.67819729161302025413774362271, 7.77197411937070164848921480044, 8.524844152988541565931793820066, 9.584354271282085285347819059871, 10.53024006592294073237424997466

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