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Properties

Label 2-525-25.21-c1-0-13
Degree $2$
Conductor $525$
Sign $0.225 - 0.974i$
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  + (−0.179 + 0.553i)2-s + (0.809 + 0.587i)3-s + (1.34 + 0.976i)4-s + (2.17 − 0.513i)5-s + (−0.471 + 0.342i)6-s − 7-s + (−1.72 + 1.25i)8-s + (0.309 + 0.951i)9-s + (−0.106 + 1.29i)10-s + (−0.434 + 1.33i)11-s + (0.513 + 1.57i)12-s + (0.370 + 1.13i)13-s + (0.179 − 0.553i)14-s + (2.06 + 0.863i)15-s + (0.642 + 1.97i)16-s + (1.94 − 1.41i)17-s + ⋯
L(s)  = 1  + (−0.127 + 0.391i)2-s + (0.467 + 0.339i)3-s + (0.671 + 0.488i)4-s + (0.973 − 0.229i)5-s + (−0.192 + 0.139i)6-s − 0.377·7-s + (−0.609 + 0.442i)8-s + (0.103 + 0.317i)9-s + (−0.0338 + 0.410i)10-s + (−0.131 + 0.403i)11-s + (0.148 + 0.456i)12-s + (0.102 + 0.316i)13-s + (0.0480 − 0.148i)14-s + (0.532 + 0.222i)15-s + (0.160 + 0.494i)16-s + (0.472 − 0.343i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.225 - 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.225 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.225 - 0.974i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (421, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.225 - 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57199 + 1.24981i\)
\(L(\frac12)\) \(\approx\) \(1.57199 + 1.24981i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 + (-2.17 + 0.513i)T \)
7 \( 1 + T \)
good2 \( 1 + (0.179 - 0.553i)T + (-1.61 - 1.17i)T^{2} \)
11 \( 1 + (0.434 - 1.33i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-0.370 - 1.13i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.94 + 1.41i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (4.65 - 3.37i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-2.63 + 8.10i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-4.58 - 3.33i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.996 - 0.723i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.24 + 6.91i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.19 + 6.75i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 1.10T + 43T^{2} \)
47 \( 1 + (1.99 + 1.45i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-8.49 - 6.17i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.57 + 4.85i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (4.56 - 14.0i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (7.84 - 5.69i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (5.43 + 3.94i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.01 + 6.20i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-1.77 - 1.28i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-10.8 + 7.85i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-5.52 + 17.0i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (11.0 + 8.03i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.64077856807805216136903132234, −10.29645163218241527316973790546, −8.984780800719580704958451992787, −8.614917211665864087425203034140, −7.35938400816226947385600979645, −6.55376975203363632175434639536, −5.65945117694393096930999108959, −4.37044309945630799394681140450, −2.98251298300946445031568924693, −2.02305546538529855382652444337, 1.30818414733691719638425669109, 2.50444024910944858045918808674, 3.35453535095487486819598149162, 5.25119959389283346331046128977, 6.24178644287862145031241938462, 6.80309555743933559348203678753, 8.023707804654433221696024372834, 9.160739705812189316599288167174, 9.839733496976924309700572706382, 10.58705415843214447455609123945

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