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Properties

Label 2-525-21.17-c1-0-16
Degree $2$
Conductor $525$
Sign $0.876 - 0.480i$
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  + (1.78 − 1.03i)2-s + (−1.08 + 1.35i)3-s + (1.12 − 1.95i)4-s + (−0.543 + 3.53i)6-s + (0.00953 + 2.64i)7-s − 0.527i·8-s + (−0.649 − 2.92i)9-s + (4.06 + 2.34i)11-s + (1.41 + 3.64i)12-s − 0.638i·13-s + (2.74 + 4.71i)14-s + (1.71 + 2.96i)16-s + (−2.07 + 3.59i)17-s + (−4.18 − 4.56i)18-s + (−0.776 + 0.448i)19-s + ⋯
L(s)  = 1  + (1.26 − 0.729i)2-s + (−0.625 + 0.779i)3-s + (0.563 − 0.976i)4-s + (−0.221 + 1.44i)6-s + (0.00360 + 0.999i)7-s − 0.186i·8-s + (−0.216 − 0.976i)9-s + (1.22 + 0.707i)11-s + (0.408 + 1.05i)12-s − 0.177i·13-s + (0.733 + 1.26i)14-s + (0.427 + 0.741i)16-s + (−0.503 + 0.871i)17-s + (−0.985 − 1.07i)18-s + (−0.178 + 0.102i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.480i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.876 - 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.876 - 0.480i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.876 - 0.480i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.23177 + 0.571318i\)
\(L(\frac12)\) \(\approx\) \(2.23177 + 0.571318i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.08 - 1.35i)T \)
5 \( 1 \)
7 \( 1 + (-0.00953 - 2.64i)T \)
good2 \( 1 + (-1.78 + 1.03i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (-4.06 - 2.34i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 0.638iT - 13T^{2} \)
17 \( 1 + (2.07 - 3.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.776 - 0.448i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.89 + 3.40i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.14iT - 29T^{2} \)
31 \( 1 + (2.02 + 1.16i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.69 + 9.85i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.10T + 41T^{2} \)
43 \( 1 + 3.14T + 43T^{2} \)
47 \( 1 + (3.40 + 5.89i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.96 - 1.13i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.254 + 0.440i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.48 + 2.59i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.41 + 4.18i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.22iT - 71T^{2} \)
73 \( 1 + (12.5 + 7.22i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.54 + 7.86i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.76T + 83T^{2} \)
89 \( 1 + (-6.90 - 11.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 12.9iT - 97T^{2} \)
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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

−11.10892336222736148168419010818, −10.46781453084265318011314522157, −9.277363006488182206240030746893, −8.645666713689402572371964469163, −6.77482357475618751352525783857, −5.92725071080385876809659286309, −5.07243402345751032745490415346, −4.23841849357771301832316620115, −3.35175267212555824705068537317, −1.95188408809953876024118276960, 1.10219595739119656779208488540, 3.20497782792408308182523701524, 4.35793837357444074152453552203, 5.20447367896774025840434668774, 6.29803299275754384613806911563, 6.86685227202534809267646362242, 7.48871091911571652784844036445, 8.788182235068920799578882448680, 10.05235744967015823124051325775, 11.35826044142029839740338557145

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