Properties

Degree $2$
Conductor $525$
Sign $1$
Motivic weight $3$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.70·2-s − 3·3-s + 14.1·4-s + 14.1·6-s − 7·7-s − 28.7·8-s + 9·9-s + 24.5·11-s − 42.3·12-s + 35.0·13-s + 32.9·14-s + 22.1·16-s + 18.4·17-s − 42.3·18-s − 67.4·19-s + 21·21-s − 115.·22-s + 145.·23-s + 86.1·24-s − 164.·26-s − 27·27-s − 98.7·28-s + 214.·29-s − 88.6·31-s + 125.·32-s − 73.7·33-s − 86.5·34-s + ⋯
L(s)  = 1  − 1.66·2-s − 0.577·3-s + 1.76·4-s + 0.959·6-s − 0.377·7-s − 1.26·8-s + 0.333·9-s + 0.674·11-s − 1.01·12-s + 0.747·13-s + 0.628·14-s + 0.345·16-s + 0.262·17-s − 0.554·18-s − 0.813·19-s + 0.218·21-s − 1.12·22-s + 1.32·23-s + 0.732·24-s − 1.24·26-s − 0.192·27-s − 0.666·28-s + 1.37·29-s − 0.513·31-s + 0.694·32-s − 0.389·33-s − 0.436·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $1$
Motivic weight: \(3\)
Character: $\chi_{525} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6284837598\)
\(L(\frac12)\) \(\approx\) \(0.6284837598\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3T \)
5 \( 1 \)
7 \( 1 + 7T \)
good2 \( 1 + 4.70T + 8T^{2} \)
11 \( 1 - 24.5T + 1.33e3T^{2} \)
13 \( 1 - 35.0T + 2.19e3T^{2} \)
17 \( 1 - 18.4T + 4.91e3T^{2} \)
19 \( 1 + 67.4T + 6.85e3T^{2} \)
23 \( 1 - 145.T + 1.21e4T^{2} \)
29 \( 1 - 214.T + 2.43e4T^{2} \)
31 \( 1 + 88.6T + 2.97e4T^{2} \)
37 \( 1 + 162.T + 5.06e4T^{2} \)
41 \( 1 + 337.T + 6.89e4T^{2} \)
43 \( 1 + 122.T + 7.95e4T^{2} \)
47 \( 1 + 354.T + 1.03e5T^{2} \)
53 \( 1 + 676.T + 1.48e5T^{2} \)
59 \( 1 - 501.T + 2.05e5T^{2} \)
61 \( 1 + 708.T + 2.26e5T^{2} \)
67 \( 1 - 907.T + 3.00e5T^{2} \)
71 \( 1 - 430.T + 3.57e5T^{2} \)
73 \( 1 + 41.3T + 3.89e5T^{2} \)
79 \( 1 - 890.T + 4.93e5T^{2} \)
83 \( 1 - 1.05e3T + 5.71e5T^{2} \)
89 \( 1 - 1.47e3T + 7.04e5T^{2} \)
97 \( 1 + 555.T + 9.12e5T^{2} \)
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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

−10.40099230716399718473465553066, −9.514644768178388124076130362418, −8.778035039137146358886626658437, −7.998355039869237451727281040568, −6.72590460896638801413599308830, −6.46481252689732640356209446685, −4.91997094614412997784028654818, −3.36926902953674733975486515684, −1.73711471114374038831705788498, −0.66305326649196143557993196086, 0.66305326649196143557993196086, 1.73711471114374038831705788498, 3.36926902953674733975486515684, 4.91997094614412997784028654818, 6.46481252689732640356209446685, 6.72590460896638801413599308830, 7.998355039869237451727281040568, 8.778035039137146358886626658437, 9.514644768178388124076130362418, 10.40099230716399718473465553066

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