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Properties

Label 2-525-21.17-c1-0-15
Degree $2$
Conductor $525$
Sign $0.417 - 0.908i$
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  + (−1.94 + 1.12i)2-s + (−0.983 + 1.42i)3-s + (1.53 − 2.65i)4-s + (0.313 − 3.88i)6-s + (1.42 + 2.22i)7-s + 2.39i·8-s + (−1.06 − 2.80i)9-s + (1.64 + 0.952i)11-s + (2.27 + 4.79i)12-s − 5.07i·13-s + (−5.29 − 2.73i)14-s + (0.369 + 0.639i)16-s + (2.22 − 3.85i)17-s + (5.22 + 4.26i)18-s + (3.85 − 2.22i)19-s + ⋯
L(s)  = 1  + (−1.37 + 0.795i)2-s + (−0.568 + 0.822i)3-s + (0.766 − 1.32i)4-s + (0.128 − 1.58i)6-s + (0.540 + 0.841i)7-s + 0.846i·8-s + (−0.354 − 0.935i)9-s + (0.497 + 0.287i)11-s + (0.656 + 1.38i)12-s − 1.40i·13-s + (−1.41 − 0.730i)14-s + (0.0923 + 0.159i)16-s + (0.540 − 0.936i)17-s + (1.23 + 1.00i)18-s + (0.884 − 0.510i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.417 - 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.417 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.417 - 0.908i$
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: \((2,\ 525,\ (\ :1/2),\ 0.417 - 0.908i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.535405 + 0.343306i\)
\(L(\frac12)\) \(\approx\) \(0.535405 + 0.343306i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.983 - 1.42i)T \)
5 \( 1 \)
7 \( 1 + (-1.42 - 2.22i)T \)
good2 \( 1 + (1.94 - 1.12i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (-1.64 - 0.952i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.07iT - 13T^{2} \)
17 \( 1 + (-2.22 + 3.85i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.85 + 2.22i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.46 + 1.42i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 8.82iT - 29T^{2} \)
31 \( 1 + (-4.81 - 2.77i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.32 - 4.02i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 0.250T + 41T^{2} \)
43 \( 1 + 9.23T + 43T^{2} \)
47 \( 1 + (-1.53 - 2.66i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.09 - 1.21i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.95 - 12.0i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.51 - 0.874i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.14 + 7.18i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.68iT - 71T^{2} \)
73 \( 1 + (-5.47 - 3.15i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.59 - 2.76i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.98T + 83T^{2} \)
89 \( 1 + (-5.43 - 9.41i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 8.94iT - 97T^{2} \)
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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

−10.69817561648495974732267647454, −9.832093900647649211474282482931, −9.347907594818333855660621744558, −8.412029456432707927050336072216, −7.64739877433720978127340869602, −6.49901970350770360074658712348, −5.62031932705079947934919514982, −4.78387787685726232541792843320, −3.01178172542952442737015194503, −0.846362969491703340207734719995, 1.09915576848393248846034261154, 1.84014065756468585223434312137, 3.55987108378251895795412279835, 5.08581937977912159110655919122, 6.50413175214204905867334044839, 7.33780160194095866286944673688, 8.092443345102824378496905701607, 8.936863481405623710394309899602, 9.960485303872413759968629679686, 10.75217303547365270711380611747

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