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Properties

Label 2-525-5.4-c3-0-8
Degree $2$
Conductor $525$
Sign $0.447 + 0.894i$
Analytic cond. $30.9760$
Root an. cond. $5.56560$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.23i·2-s + 3i·3-s − 9.94·4-s − 12.7·6-s + 7i·7-s − 8.23i·8-s − 9·9-s − 41.5·11-s − 29.8i·12-s + 88.9i·13-s − 29.6·14-s − 44.6·16-s + 120. i·17-s − 38.1i·18-s + 112.·19-s + ⋯
L(s)  = 1  + 1.49i·2-s + 0.577i·3-s − 1.24·4-s − 0.864·6-s + 0.377i·7-s − 0.363i·8-s − 0.333·9-s − 1.13·11-s − 0.717i·12-s + 1.89i·13-s − 0.566·14-s − 0.697·16-s + 1.71i·17-s − 0.499i·18-s + 1.35·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(30.9760\)
Root analytic conductor: \(5.56560\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :3/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9432985033\)
\(L(\frac12)\) \(\approx\) \(0.9432985033\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3iT \)
5 \( 1 \)
7 \( 1 - 7iT \)
good2 \( 1 - 4.23iT - 8T^{2} \)
11 \( 1 + 41.5T + 1.33e3T^{2} \)
13 \( 1 - 88.9iT - 2.19e3T^{2} \)
17 \( 1 - 120. iT - 4.91e3T^{2} \)
19 \( 1 - 112.T + 6.85e3T^{2} \)
23 \( 1 + 115. iT - 1.21e4T^{2} \)
29 \( 1 - 144.T + 2.43e4T^{2} \)
31 \( 1 + 258.T + 2.97e4T^{2} \)
37 \( 1 + 48.3iT - 5.06e4T^{2} \)
41 \( 1 - 200.T + 6.89e4T^{2} \)
43 \( 1 + 218. iT - 7.95e4T^{2} \)
47 \( 1 + 575. iT - 1.03e5T^{2} \)
53 \( 1 + 184. iT - 1.48e5T^{2} \)
59 \( 1 - 151.T + 2.05e5T^{2} \)
61 \( 1 + 529.T + 2.26e5T^{2} \)
67 \( 1 + 1.28iT - 3.00e5T^{2} \)
71 \( 1 + 61.4T + 3.57e5T^{2} \)
73 \( 1 - 484. iT - 3.89e5T^{2} \)
79 \( 1 + 878.T + 4.93e5T^{2} \)
83 \( 1 - 491. iT - 5.71e5T^{2} \)
89 \( 1 - 415.T + 7.04e5T^{2} \)
97 \( 1 - 1.03e3iT - 9.12e5T^{2} \)
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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

−11.04569102467808904206876435311, −10.10178087538070155270015979860, −9.019983068034292439466931812387, −8.505025928829268358651699910681, −7.50474014685843443155313603229, −6.59731488510146135835771490082, −5.70485548349316412714677842163, −4.90853104921410795360070234082, −3.88900486469867854799395718820, −2.18498110468207537380224385567, 0.30721832406503930222481964623, 1.21100416868405288663222589763, 2.83528033427096463647666197159, 3.13253116104481272972588001508, 4.83976750361355594094720533691, 5.68003048887681946222963262919, 7.38996629216998087385113079209, 7.76712190599655738149455224128, 9.215197978436076251178106851999, 9.948035445078937996404182533189

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