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Properties

Label 2-525-25.6-c1-0-2
Degree $2$
Conductor $525$
Sign $0.837 + 0.546i$
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  + (−0.612 − 1.88i)2-s + (−0.809 + 0.587i)3-s + (−1.55 + 1.12i)4-s + (−2.21 + 0.335i)5-s + (1.60 + 1.16i)6-s + 7-s + (−0.124 − 0.0904i)8-s + (0.309 − 0.951i)9-s + (1.98 + 3.95i)10-s + (0.881 + 2.71i)11-s + (0.594 − 1.82i)12-s + (−0.350 + 1.07i)13-s + (−0.612 − 1.88i)14-s + (1.59 − 1.57i)15-s + (−1.28 + 3.94i)16-s + (2.41 + 1.75i)17-s + ⋯
L(s)  = 1  + (−0.432 − 1.33i)2-s + (−0.467 + 0.339i)3-s + (−0.777 + 0.564i)4-s + (−0.988 + 0.150i)5-s + (0.654 + 0.475i)6-s + 0.377·7-s + (−0.0439 − 0.0319i)8-s + (0.103 − 0.317i)9-s + (0.627 + 1.25i)10-s + (0.265 + 0.817i)11-s + (0.171 − 0.527i)12-s + (−0.0972 + 0.299i)13-s + (−0.163 − 0.503i)14-s + (0.410 − 0.405i)15-s + (−0.320 + 0.986i)16-s + (0.585 + 0.425i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.837 + 0.546i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.837 + 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.837 + 0.546i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.837 + 0.546i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.738503 - 0.219723i\)
\(L(\frac12)\) \(\approx\) \(0.738503 - 0.219723i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.21 - 0.335i)T \)
7 \( 1 - T \)
good2 \( 1 + (0.612 + 1.88i)T + (-1.61 + 1.17i)T^{2} \)
11 \( 1 + (-0.881 - 2.71i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (0.350 - 1.07i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-2.41 - 1.75i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.80 - 2.04i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (1.55 + 4.77i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-2.78 + 2.02i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-6.54 - 4.75i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.435 + 1.34i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.479 + 1.47i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 10.2T + 43T^{2} \)
47 \( 1 + (-0.590 + 0.429i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (6.44 - 4.67i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.09 - 6.43i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-4.53 - 13.9i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (4.17 + 3.03i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (4.85 - 3.52i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.43 + 4.42i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (2.04 - 1.48i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-6.21 - 4.51i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-1.27 - 3.90i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (1.95 - 1.41i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.65505734288249037421960923138, −10.28858790274810712799983543462, −9.283337602445035618521710101056, −8.350008930557493105282876001350, −7.32338834728285326380523319951, −6.16567541928743763864573489340, −4.61632194518066416602559970824, −3.92816953026831119354880496964, −2.69465841523394244022370398681, −1.11437220514764259761700153137, 0.71224911701561195357589580784, 3.15354008342203296480264756478, 4.69616433877570645390069717995, 5.59046895197991892834633038541, 6.49801653163889048975380610248, 7.53120385027494953767028345653, 7.904662762047720898356175101592, 8.789623727855336767192925654185, 9.792685388620194578140124226999, 11.22529264460085235390305734582

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