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Properties

Label 2-525-3.2-c2-0-31
Degree $2$
Conductor $525$
Sign $-0.274 + 0.961i$
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  − 3.50i·2-s + (−0.822 + 2.88i)3-s − 8.29·4-s + (10.1 + 2.88i)6-s − 2.64·7-s + 15.0i·8-s + (−7.64 − 4.74i)9-s + 7.01i·11-s + (6.82 − 23.9i)12-s + 11.6·13-s + 9.27i·14-s + 19.5·16-s + 4.52i·17-s + (−16.6 + 26.8i)18-s + 16.2·19-s + ⋯
L(s)  = 1  − 1.75i·2-s + (−0.274 + 0.961i)3-s − 2.07·4-s + (1.68 + 0.480i)6-s − 0.377·7-s + 1.88i·8-s + (−0.849 − 0.527i)9-s + 0.637i·11-s + (0.568 − 1.99i)12-s + 0.895·13-s + 0.662i·14-s + 1.22·16-s + 0.266i·17-s + (−0.924 + 1.48i)18-s + 0.854·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.274 + 0.961i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.274 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.274 + 0.961i$
Analytic conductor: \(14.3052\)
Root analytic conductor: \(3.78222\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (176, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1),\ -0.274 + 0.961i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.225127393\)
\(L(\frac12)\) \(\approx\) \(1.225127393\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.822 - 2.88i)T \)
5 \( 1 \)
7 \( 1 + 2.64T \)
good2 \( 1 + 3.50iT - 4T^{2} \)
11 \( 1 - 7.01iT - 121T^{2} \)
13 \( 1 - 11.6T + 169T^{2} \)
17 \( 1 - 4.52iT - 289T^{2} \)
19 \( 1 - 16.2T + 361T^{2} \)
23 \( 1 + 25.5iT - 529T^{2} \)
29 \( 1 + 9.49iT - 841T^{2} \)
31 \( 1 - 28.7T + 961T^{2} \)
37 \( 1 - 33.0T + 1.36e3T^{2} \)
41 \( 1 + 67.1iT - 1.68e3T^{2} \)
43 \( 1 - 24.1T + 1.84e3T^{2} \)
47 \( 1 + 33.0iT - 2.20e3T^{2} \)
53 \( 1 + 15.1iT - 2.80e3T^{2} \)
59 \( 1 + 92.3iT - 3.48e3T^{2} \)
61 \( 1 + 57.5T + 3.72e3T^{2} \)
67 \( 1 + 15.1T + 4.48e3T^{2} \)
71 \( 1 - 70.5iT - 5.04e3T^{2} \)
73 \( 1 - 76.7T + 5.32e3T^{2} \)
79 \( 1 - 127.T + 6.24e3T^{2} \)
83 \( 1 - 74.2iT - 6.88e3T^{2} \)
89 \( 1 - 127. iT - 7.92e3T^{2} \)
97 \( 1 - 23.1T + 9.40e3T^{2} \)
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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

−10.51359807262652186839299252273, −9.774520972518676975598907669473, −9.158567756638184695096330006154, −8.227960019350486066082398105646, −6.47286088224640750201629114869, −5.23032445799730882626244076864, −4.24120229983762025144076543010, −3.50467545047967323818612310822, −2.39305564294611814956463855922, −0.70147442027864791840994660208, 0.963362686823475302955792537051, 3.19378917865023948813616325932, 4.76333628650102441789081302355, 5.92479347971712120840053558528, 6.19141050016745642068693710342, 7.30495068543521718695131673484, 7.88115952816896977578748892777, 8.746832686995997591853325358165, 9.574753901328607572081552493785, 11.01803028316800511784575460114

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