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Properties

Label 2-525-105.89-c1-0-18
Degree $2$
Conductor $525$
Sign $0.0375 - 0.999i$
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  + (−1.03 + 1.78i)2-s + (−1.35 + 1.08i)3-s + (−1.12 − 1.95i)4-s + (−0.543 − 3.53i)6-s + (2.64 + 0.00953i)7-s + 0.527·8-s + (0.649 − 2.92i)9-s + (4.06 − 2.34i)11-s + (3.64 + 1.41i)12-s + 0.638·13-s + (−2.74 + 4.71i)14-s + (1.71 − 2.96i)16-s + (3.59 − 2.07i)17-s + (4.56 + 4.18i)18-s + (0.776 + 0.448i)19-s + ⋯
L(s)  = 1  + (−0.729 + 1.26i)2-s + (−0.779 + 0.625i)3-s + (−0.563 − 0.976i)4-s + (−0.221 − 1.44i)6-s + (0.999 + 0.00360i)7-s + 0.186·8-s + (0.216 − 0.976i)9-s + (1.22 − 0.707i)11-s + (1.05 + 0.408i)12-s + 0.177·13-s + (−0.733 + 1.26i)14-s + (0.427 − 0.741i)16-s + (0.871 − 0.503i)17-s + (1.07 + 0.985i)18-s + (0.178 + 0.102i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0375 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0375 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.0375 - 0.999i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.0375 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.634133 + 0.610743i\)
\(L(\frac12)\) \(\approx\) \(0.634133 + 0.610743i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.35 - 1.08i)T \)
5 \( 1 \)
7 \( 1 + (-2.64 - 0.00953i)T \)
good2 \( 1 + (1.03 - 1.78i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (-4.06 + 2.34i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 0.638T + 13T^{2} \)
17 \( 1 + (-3.59 + 2.07i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.776 - 0.448i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.40 + 5.89i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.14iT - 29T^{2} \)
31 \( 1 + (2.02 - 1.16i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (9.85 + 5.69i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.10T + 41T^{2} \)
43 \( 1 - 3.14iT - 43T^{2} \)
47 \( 1 + (5.89 + 3.40i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.13 + 1.96i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.254 + 0.440i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.48 - 2.59i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.18 - 2.41i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.22iT - 71T^{2} \)
73 \( 1 + (-7.22 - 12.5i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.54 + 7.86i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.76iT - 83T^{2} \)
89 \( 1 + (6.90 - 11.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 12.9T + 97T^{2} \)
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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

−10.98623013973110265852817728395, −10.01716440449448197135477843914, −9.007236213846452091422258163301, −8.554919731603835429336030276690, −7.34851218733007907901158906453, −6.56548215514098158413238381521, −5.65361118308273213718608970227, −4.88797098335383973556676219476, −3.54645301302279903708969256578, −0.953676219199042195452751903392, 1.22256171587089618153935878113, 1.87460779653704897349941783922, 3.57972826132597525387944986371, 4.89244709534078566551977702354, 6.02598581494046878226575144229, 7.20219111409326551107560489667, 8.084860481838200375675415654726, 9.085657612896252861503532686276, 9.975360746833921984802521783109, 10.82770907587846766046226794812

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