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Properties

Label 2-525-35.12-c1-0-12
Degree $2$
Conductor $525$
Sign $0.296 + 0.955i$
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  + (−2.01 + 0.538i)2-s + (−0.258 + 0.965i)3-s + (2.02 − 1.16i)4-s − 2.08i·6-s + (−1.99 + 1.74i)7-s + (−0.495 + 0.495i)8-s + (−0.866 − 0.499i)9-s + (−0.971 − 1.68i)11-s + (0.604 + 2.25i)12-s + (1.84 + 1.84i)13-s + (3.06 − 4.57i)14-s + (−1.60 + 2.78i)16-s + (−6.06 − 1.62i)17-s + (2.01 + 0.538i)18-s + (1.50 − 2.61i)19-s + ⋯
L(s)  = 1  + (−1.42 + 0.381i)2-s + (−0.149 + 0.557i)3-s + (1.01 − 0.584i)4-s − 0.850i·6-s + (−0.752 + 0.658i)7-s + (−0.175 + 0.175i)8-s + (−0.288 − 0.166i)9-s + (−0.292 − 0.507i)11-s + (0.174 + 0.651i)12-s + (0.511 + 0.511i)13-s + (0.819 − 1.22i)14-s + (−0.401 + 0.695i)16-s + (−1.47 − 0.394i)17-s + (0.474 + 0.127i)18-s + (0.345 − 0.599i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.296 + 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.296 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.296 + 0.955i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.296 + 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.171202 - 0.126167i\)
\(L(\frac12)\) \(\approx\) \(0.171202 - 0.126167i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.258 - 0.965i)T \)
5 \( 1 \)
7 \( 1 + (1.99 - 1.74i)T \)
good2 \( 1 + (2.01 - 0.538i)T + (1.73 - i)T^{2} \)
11 \( 1 + (0.971 + 1.68i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.84 - 1.84i)T + 13iT^{2} \)
17 \( 1 + (6.06 + 1.62i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.50 + 2.61i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.43 + 5.35i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 5.27iT - 29T^{2} \)
31 \( 1 + (5.10 - 2.94i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.901 + 0.241i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 10.2iT - 41T^{2} \)
43 \( 1 + (-6.54 + 6.54i)T - 43iT^{2} \)
47 \( 1 + (1.00 + 3.75i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.12 + 0.301i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (3.06 + 5.30i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.50 - 3.75i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.85 + 14.3i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 15.1T + 71T^{2} \)
73 \( 1 + (-1.62 + 6.05i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (14.2 + 8.21i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.197 + 0.197i)T + 83iT^{2} \)
89 \( 1 + (6.08 - 10.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (10.9 - 10.9i)T - 97iT^{2} \)
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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

−10.68159375016741588958024480048, −9.435774054986685063430159937006, −9.000855562156338603348450972836, −8.423727764585154680807625158275, −7.02162149249535378038886741751, −6.44035870284485057202529347311, −5.23894171536763264941660822561, −3.83111405905507181777000520073, −2.32279525817533784589112354991, −0.21267031080954465330630929355, 1.32141489936536925715217679328, 2.63665851609018384204209048515, 4.15046695830682241042928957466, 5.79270416655537619318082372011, 6.83580134355934076148255560383, 7.67040618658614735751769374870, 8.319317348742332413514238073842, 9.504384920419469532601905727771, 9.935311939835367030311450317323, 11.00847580107991838492583295637

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