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Properties

Label 2-525-5.4-c3-0-3
Degree $2$
Conductor $525$
Sign $-0.447 - 0.894i$
Analytic cond. $30.9760$
Root an. cond. $5.56560$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.27i·2-s + 3i·3-s + 2.82·4-s + 6.82·6-s − 7i·7-s − 24.6i·8-s − 9·9-s − 40.7·11-s + 8.47i·12-s + 53.2i·13-s − 15.9·14-s − 33.4·16-s − 4.54i·17-s + 20.4i·18-s − 122.·19-s + ⋯
L(s)  = 1  − 0.804i·2-s + 0.577i·3-s + 0.353·4-s + 0.464·6-s − 0.377i·7-s − 1.08i·8-s − 0.333·9-s − 1.11·11-s + 0.203i·12-s + 1.13i·13-s − 0.303·14-s − 0.522·16-s − 0.0649i·17-s + 0.268i·18-s − 1.48·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(30.9760\)
Root analytic conductor: \(5.56560\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :3/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4697585725\)
\(L(\frac12)\) \(\approx\) \(0.4697585725\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3iT \)
5 \( 1 \)
7 \( 1 + 7iT \)
good2 \( 1 + 2.27iT - 8T^{2} \)
11 \( 1 + 40.7T + 1.33e3T^{2} \)
13 \( 1 - 53.2iT - 2.19e3T^{2} \)
17 \( 1 + 4.54iT - 4.91e3T^{2} \)
19 \( 1 + 122.T + 6.85e3T^{2} \)
23 \( 1 - 131. iT - 1.21e4T^{2} \)
29 \( 1 - 216.T + 2.43e4T^{2} \)
31 \( 1 + 251.T + 2.97e4T^{2} \)
37 \( 1 + 11.8iT - 5.06e4T^{2} \)
41 \( 1 + 111.T + 6.89e4T^{2} \)
43 \( 1 - 369. iT - 7.95e4T^{2} \)
47 \( 1 - 262. iT - 1.03e5T^{2} \)
53 \( 1 + 567. iT - 1.48e5T^{2} \)
59 \( 1 + 839.T + 2.05e5T^{2} \)
61 \( 1 + 485.T + 2.26e5T^{2} \)
67 \( 1 - 333. iT - 3.00e5T^{2} \)
71 \( 1 - 590.T + 3.57e5T^{2} \)
73 \( 1 - 490. iT - 3.89e5T^{2} \)
79 \( 1 + 121.T + 4.93e5T^{2} \)
83 \( 1 - 609. iT - 5.71e5T^{2} \)
89 \( 1 + 719.T + 7.04e5T^{2} \)
97 \( 1 - 637. iT - 9.12e5T^{2} \)
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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

−10.88095074123170531887745704072, −10.03149498047800209441346361617, −9.311409401572692460617166551207, −8.147754310579269196974763621439, −7.08697797149362872071885649091, −6.15319120093791774961963025557, −4.80450666101873344190681166910, −3.86996060890320856476038129751, −2.77815353232561105059412950443, −1.65657257572099203224233433504, 0.12620848996950542813967775816, 2.07164878521730729187503599492, 2.94947668077506876579627308906, 4.83518221657139467560580866086, 5.75704740163529757987255810300, 6.47519899837809723402562832447, 7.46029527042246153351728887922, 8.189219714961483433323727455754, 8.799660926272652539991113508700, 10.59467336469439329448886081128

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