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Properties

Label 2-525-35.33-c1-0-8
Degree $2$
Conductor $525$
Sign $-0.808 - 0.589i$
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  + (0.582 + 2.17i)2-s + (0.965 + 0.258i)3-s + (−2.65 + 1.53i)4-s + 2.25i·6-s + (2.56 − 0.660i)7-s + (−1.69 − 1.69i)8-s + (0.866 + 0.499i)9-s + (0.329 + 0.571i)11-s + (−2.96 + 0.793i)12-s + (−2.55 + 2.55i)13-s + (2.92 + 5.18i)14-s + (−0.366 + 0.635i)16-s + (−1.45 + 5.43i)17-s + (−0.582 + 2.17i)18-s + (1.48 − 2.56i)19-s + ⋯
L(s)  = 1  + (0.411 + 1.53i)2-s + (0.557 + 0.149i)3-s + (−1.32 + 0.766i)4-s + 0.918i·6-s + (0.968 − 0.249i)7-s + (−0.599 − 0.599i)8-s + (0.288 + 0.166i)9-s + (0.0994 + 0.172i)11-s + (−0.854 + 0.229i)12-s + (−0.709 + 0.709i)13-s + (0.782 + 1.38i)14-s + (−0.0916 + 0.158i)16-s + (−0.353 + 1.31i)17-s + (−0.137 + 0.512i)18-s + (0.339 − 0.588i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.808 - 0.589i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.808 - 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.808 - 0.589i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (418, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.808 - 0.589i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.672373 + 2.06354i\)
\(L(\frac12)\) \(\approx\) \(0.672373 + 2.06354i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.965 - 0.258i)T \)
5 \( 1 \)
7 \( 1 + (-2.56 + 0.660i)T \)
good2 \( 1 + (-0.582 - 2.17i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (-0.329 - 0.571i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.55 - 2.55i)T - 13iT^{2} \)
17 \( 1 + (1.45 - 5.43i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.48 + 2.56i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.271 + 0.0726i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 5.03iT - 29T^{2} \)
31 \( 1 + (-6.53 + 3.77i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.08 + 7.79i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 7.07iT - 41T^{2} \)
43 \( 1 + (8.53 + 8.53i)T + 43iT^{2} \)
47 \( 1 + (-11.6 + 3.11i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (1.24 - 4.65i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-0.782 - 1.35i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.02 + 0.589i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.84 - 0.762i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 3.13T + 71T^{2} \)
73 \( 1 + (1.55 + 0.417i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (6.17 + 3.56i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.14 - 2.14i)T - 83iT^{2} \)
89 \( 1 + (-2.24 + 3.88i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.33 + 3.33i)T + 97iT^{2} \)
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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

−11.21504043455977403804331341554, −10.13922495316594319472802389238, −9.003787594750533151496752055595, −8.302859144667857400257464146102, −7.53777277904267292339379390710, −6.80173605287793671634119239186, −5.70645813444276095873263152457, −4.62514265432281641459567450182, −4.05571635223199938219803913781, −2.10298723817911591358181709511, 1.21722768984561542485850139029, 2.47504761951082295670349447067, 3.27316259145523980392204643855, 4.61124042827097731745985952457, 5.26616917925123560457129984087, 7.00963059969440855652682988335, 8.037848783500449282657228709790, 8.968081765642887232923557155089, 9.877193290927337978511614428857, 10.59125962844023352917225582844

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