Properties

Label 2-525-21.17-c1-0-25
Degree $2$
Conductor $525$
Sign $-0.205 + 0.978i$
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  + (−1.5 − 0.866i)3-s + (−1 + 1.73i)4-s + (−2 + 1.73i)7-s + (1.5 + 2.59i)9-s + (3 − 1.73i)12-s − 6.92i·13-s + (−1.99 − 3.46i)16-s + (3 − 1.73i)19-s + (4.5 − 0.866i)21-s − 5.19i·27-s + (−0.999 − 5.19i)28-s + (−7.5 − 4.33i)31-s − 6·36-s + (−5.5 − 9.52i)37-s + (−5.99 + 10.3i)39-s + ⋯
L(s)  = 1  + (−0.866 − 0.499i)3-s + (−0.5 + 0.866i)4-s + (−0.755 + 0.654i)7-s + (0.5 + 0.866i)9-s + (0.866 − 0.499i)12-s − 1.92i·13-s + (−0.499 − 0.866i)16-s + (0.688 − 0.397i)19-s + (0.981 − 0.188i)21-s − 0.999i·27-s + (−0.188 − 0.981i)28-s + (−1.34 − 0.777i)31-s − 36-s + (−0.904 − 1.56i)37-s + (−0.960 + 1.66i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.205 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.205 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.205 + 0.978i$
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: \((2,\ 525,\ (\ :1/2),\ -0.205 + 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.295113 - 0.363424i\)
\(L(\frac12)\) \(\approx\) \(0.295113 - 0.363424i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 + 0.866i)T \)
5 \( 1 \)
7 \( 1 + (2 - 1.73i)T \)
good2 \( 1 + (1 - 1.73i)T^{2} \)
11 \( 1 + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 6.92iT - 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3 + 1.73i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (7.5 + 4.33i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.5 + 9.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 5T + 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-13.5 + 7.79i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (8 - 13.8i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-1.5 - 0.866i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.5 + 14.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 5.19iT - 97T^{2} \)
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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

−10.70502872866695506007782725938, −9.735795588185913057371186093046, −8.773706315617616460455465209010, −7.75869065758754155126310766525, −7.11471536086813389371964565444, −5.77625279446492408405608457716, −5.24466650318102254395579568711, −3.72688113226704224038148664973, −2.60421819551448974455189700880, −0.33065262391938944937574170585, 1.39515444855475773652534463476, 3.69431087386852630741316490631, 4.51713878205467890193379286635, 5.48896659900045136605971722695, 6.49284029989436373858784559154, 7.08010138820884867212818598528, 8.875271833987165040728417833889, 9.549037082288771884611042367968, 10.16439879762252641259284961504, 10.98388937734398707745362662684

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