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Properties

Label 2-525-105.59-c1-0-29
Degree $2$
Conductor $525$
Sign $-0.456 + 0.889i$
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  + (−0.192 − 0.334i)2-s + (−1.72 − 0.139i)3-s + (0.925 − 1.60i)4-s + (0.286 + 0.603i)6-s + (1.17 − 2.36i)7-s − 1.48·8-s + (2.96 + 0.480i)9-s + (2.20 + 1.27i)11-s + (−1.82 + 2.63i)12-s + 3.06·13-s + (−1.01 + 0.0640i)14-s + (−1.56 − 2.71i)16-s + (−5.59 − 3.23i)17-s + (−0.410 − 1.08i)18-s + (1.03 − 0.597i)19-s + ⋯
L(s)  = 1  + (−0.136 − 0.236i)2-s + (−0.996 − 0.0803i)3-s + (0.462 − 0.801i)4-s + (0.116 + 0.246i)6-s + (0.444 − 0.895i)7-s − 0.525·8-s + (0.987 + 0.160i)9-s + (0.663 + 0.383i)11-s + (−0.525 + 0.761i)12-s + 0.850·13-s + (−0.272 + 0.0171i)14-s + (−0.391 − 0.677i)16-s + (−1.35 − 0.783i)17-s + (−0.0968 − 0.254i)18-s + (0.237 − 0.137i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.456 + 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.456 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.456 + 0.889i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (374, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.456 + 0.889i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.549260 - 0.899532i\)
\(L(\frac12)\) \(\approx\) \(0.549260 - 0.899532i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.72 + 0.139i)T \)
5 \( 1 \)
7 \( 1 + (-1.17 + 2.36i)T \)
good2 \( 1 + (0.192 + 0.334i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (-2.20 - 1.27i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.06T + 13T^{2} \)
17 \( 1 + (5.59 + 3.23i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.03 + 0.597i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.52 - 2.64i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 7.77iT - 29T^{2} \)
31 \( 1 + (5.95 + 3.43i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.07 + 1.77i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.31T + 41T^{2} \)
43 \( 1 - 5.46iT - 43T^{2} \)
47 \( 1 + (-2.78 + 1.61i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.62 + 11.4i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.98 - 3.43i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.08 - 4.67i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.04 + 1.75i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.921iT - 71T^{2} \)
73 \( 1 + (0.148 - 0.256i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.14 - 7.18i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.11iT - 83T^{2} \)
89 \( 1 + (-9.41 - 16.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 12.3T + 97T^{2} \)
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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

−10.80309691498616352498045450802, −9.887610503986167245275633573477, −9.124748994978642349995690470239, −7.55453560533062952770136453993, −6.79405766783902641603020101975, −6.04065417766205428894272667689, −4.94641346911443748024874840716, −3.99737711965804395055666510427, −1.96652185548473586944913405548, −0.76047967411758151567930817493, 1.77022696237787590571961104531, 3.42599643048289697853830806970, 4.57878039390458556298395653660, 5.85608174010314297571310429004, 6.46574432511998157407551606029, 7.38483457280132196956165649911, 8.707742248389006770644075775198, 8.968666044752562522580323749836, 10.73111543873379975315893367250, 11.08468550461463624913782717889

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