Properties

Label 2-525-15.14-c2-0-60
Degree $2$
Conductor $525$
Sign $0.982 - 0.184i$
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  + 3.50·2-s + (2.88 + 0.822i)3-s + 8.29·4-s + (10.1 + 2.88i)6-s − 2.64i·7-s + 15.0·8-s + (7.64 + 4.74i)9-s + 7.01i·11-s + (23.9 + 6.82i)12-s − 11.6i·13-s − 9.27i·14-s + 19.5·16-s − 4.52·17-s + (26.8 + 16.6i)18-s − 16.2·19-s + ⋯
L(s)  = 1  + 1.75·2-s + (0.961 + 0.274i)3-s + 2.07·4-s + (1.68 + 0.480i)6-s − 0.377i·7-s + 1.88·8-s + (0.849 + 0.527i)9-s + 0.637i·11-s + (1.99 + 0.568i)12-s − 0.895i·13-s − 0.662i·14-s + 1.22·16-s − 0.266·17-s + (1.48 + 0.924i)18-s − 0.854·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.184i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.982 - 0.184i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.982 - 0.184i$
Analytic conductor: \(14.3052\)
Root analytic conductor: \(3.78222\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1),\ 0.982 - 0.184i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(6.577007349\)
\(L(\frac12)\) \(\approx\) \(6.577007349\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

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

−10.63829335365080442465274327272, −10.16345578403717864350414893598, −8.786171821851555102855031381019, −7.72287694054385483488468212175, −6.91223462741953654407127595359, −5.81601820711913577837744587248, −4.65855108387850704520744380967, −4.03165429706993927815201551912, −3.00389501577775449227561414677, −1.99517781029072979022218345859, 1.91533682047349905411500285524, 2.82933241469229204026137297807, 3.91191534894362798689482651886, 4.64060178746772001027237691033, 6.06865492446162727855660998710, 6.57844418286249444349198339984, 7.76682000891911080658484193614, 8.692332358840551650563794259550, 9.756881631522756612302852332176, 11.01764801593169216442569315981

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