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Properties

Label 2-525-35.34-c2-0-13
Degree $2$
Conductor $525$
Sign $-0.725 - 0.688i$
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 − 1.73·3-s − 8.27·4-s − 6.06i·6-s + (2.03 − 6.69i)7-s − 14.9i·8-s + 2.99·9-s − 2.03·11-s + 14.3·12-s + 18.0·13-s + (23.4 + 7.13i)14-s + 19.3·16-s + 1.07·17-s + 10.5i·18-s + 28.7i·19-s + ⋯
L(s)  = 1  + 1.75i·2-s − 0.577·3-s − 2.06·4-s − 1.01i·6-s + (0.290 − 0.956i)7-s − 1.87i·8-s + 0.333·9-s − 0.184·11-s + 1.19·12-s + 1.38·13-s + (1.67 + 0.509i)14-s + 1.21·16-s + 0.0631·17-s + 0.583i·18-s + 1.51i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.725 - 0.688i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.725 - 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.725 - 0.688i$
Analytic conductor: \(14.3052\)
Root analytic conductor: \(3.78222\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1),\ -0.725 - 0.688i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.296560902\)
\(L(\frac12)\) \(\approx\) \(1.296560902\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.73T \)
5 \( 1 \)
7 \( 1 + (-2.03 + 6.69i)T \)
good2 \( 1 - 3.50iT - 4T^{2} \)
11 \( 1 + 2.03T + 121T^{2} \)
13 \( 1 - 18.0T + 169T^{2} \)
17 \( 1 - 1.07T + 289T^{2} \)
19 \( 1 - 28.7iT - 361T^{2} \)
23 \( 1 + 24.8iT - 529T^{2} \)
29 \( 1 - 38.4T + 841T^{2} \)
31 \( 1 - 44.0iT - 961T^{2} \)
37 \( 1 - 37.2iT - 1.36e3T^{2} \)
41 \( 1 + 49.9iT - 1.68e3T^{2} \)
43 \( 1 - 9.58iT - 1.84e3T^{2} \)
47 \( 1 - 55.6T + 2.20e3T^{2} \)
53 \( 1 + 57.4iT - 2.80e3T^{2} \)
59 \( 1 - 101. iT - 3.48e3T^{2} \)
61 \( 1 - 31.9iT - 3.72e3T^{2} \)
67 \( 1 - 95.7iT - 4.48e3T^{2} \)
71 \( 1 + 25.8T + 5.04e3T^{2} \)
73 \( 1 - 95.6T + 5.32e3T^{2} \)
79 \( 1 + 28.1T + 6.24e3T^{2} \)
83 \( 1 - 103.T + 6.88e3T^{2} \)
89 \( 1 - 29.3iT - 7.92e3T^{2} \)
97 \( 1 + 67.6T + 9.40e3T^{2} \)
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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

−10.64505719153089991883082886113, −10.17159990507378307460142192844, −8.683980731722789332827786593955, −8.201278869784030652535217770362, −7.16935428773575574382262936127, −6.46314671836481274023245734307, −5.68438992157462119705723184239, −4.66130618071667095948598196284, −3.78394031236116701672372464994, −1.01157664527650187362115619338, 0.76700015672777063990942246457, 2.06390022037017765412413895568, 3.17600952598568824539740162108, 4.36409337750959823110408469445, 5.31310494389339299608812376152, 6.35371672349569739657958735203, 7.978777949381844482403174586363, 9.012102087517156181511552703275, 9.538029360954744823869207059743, 10.73746706635134465759619004971

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