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Properties

Label 2-525-35.19-c2-0-23
Degree $2$
Conductor $525$
Sign $0.287 - 0.957i$
Analytic cond. $14.3052$
Root an. cond. $3.78222$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.44 + 0.836i)2-s + (0.866 + 1.5i)3-s + (−0.599 − 1.03i)4-s + 2.89i·6-s + (5.13 + 4.76i)7-s − 8.70i·8-s + (−1.5 + 2.59i)9-s + (6.91 + 11.9i)11-s + (1.03 − 1.79i)12-s − 6.12·13-s + (3.45 + 11.1i)14-s + (4.88 − 8.45i)16-s + (−1.23 − 2.14i)17-s + (−4.34 + 2.51i)18-s + (24.2 + 13.9i)19-s + ⋯
L(s)  = 1  + (0.724 + 0.418i)2-s + (0.288 + 0.5i)3-s + (−0.149 − 0.259i)4-s + 0.483i·6-s + (0.733 + 0.680i)7-s − 1.08i·8-s + (−0.166 + 0.288i)9-s + (0.628 + 1.08i)11-s + (0.0865 − 0.149i)12-s − 0.470·13-s + (0.246 + 0.799i)14-s + (0.305 − 0.528i)16-s + (−0.0729 − 0.126i)17-s + (−0.241 + 0.139i)18-s + (1.27 + 0.736i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.287 - 0.957i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.287 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.287 - 0.957i$
Analytic conductor: \(14.3052\)
Root analytic conductor: \(3.78222\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1),\ 0.287 - 0.957i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.907626079\)
\(L(\frac12)\) \(\approx\) \(2.907626079\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 - 1.5i)T \)
5 \( 1 \)
7 \( 1 + (-5.13 - 4.76i)T \)
good2 \( 1 + (-1.44 - 0.836i)T + (2 + 3.46i)T^{2} \)
11 \( 1 + (-6.91 - 11.9i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 6.12T + 169T^{2} \)
17 \( 1 + (1.23 + 2.14i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-24.2 - 13.9i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-11.4 - 6.62i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 27.6T + 841T^{2} \)
31 \( 1 + (16.2 - 9.36i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-35.5 - 20.5i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 22.5iT - 1.68e3T^{2} \)
43 \( 1 + 7.60iT - 1.84e3T^{2} \)
47 \( 1 + (-6.88 + 11.9i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (80.1 - 46.2i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-61.5 + 35.5i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (100. + 57.8i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (9.87 - 5.70i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 99.4T + 5.04e3T^{2} \)
73 \( 1 + (52.0 + 90.1i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-64.4 + 111. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 30.3T + 6.88e3T^{2} \)
89 \( 1 + (93.9 + 54.2i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 153.T + 9.40e3T^{2} \)
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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

−10.78940071256740659369552583764, −9.628585085128533859883128599843, −9.393371802804256489706517199770, −8.062704239829070979514758498915, −7.11336070661905553597084866786, −5.98784996131486666738139278943, −4.98893735487945260394403604650, −4.50189467219890374115104302232, −3.17614589519406964649517832902, −1.56198481222479898834227497044, 1.00873562666584508818126410010, 2.58744330396297700953304484832, 3.58938624934340226142691011495, 4.59601111619695315562207986713, 5.59077495974716358480285862464, 6.91139851596920584944172790761, 7.81190956604292535929297360093, 8.560375753100634939551750124217, 9.482188734043659845461768860822, 10.94418313776519042502348806870

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