Properties

Label 2-525-15.8-c1-0-34
Degree $2$
Conductor $525$
Sign $-0.991 - 0.129i$
Analytic cond. $4.19214$
Root an. cond. $2.04747$
Motivic weight $1$
Arithmetic yes
Rational no
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

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.991 - 0.129i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (218, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.991 - 0.129i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0578050 + 0.889395i\)
\(L(\frac12)\) \(\approx\) \(0.0578050 + 0.889395i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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