Properties

Label 2-3525-1.1-c1-0-117
Degree $2$
Conductor $3525$
Sign $1$
Analytic cond. $28.1472$
Root an. cond. $5.30539$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.62·2-s − 3-s + 4.89·4-s − 2.62·6-s + 5.24·7-s + 7.58·8-s + 9-s + 2.45·11-s − 4.89·12-s − 1.47·13-s + 13.7·14-s + 10.1·16-s + 4.14·17-s + 2.62·18-s + 4.25·19-s − 5.24·21-s + 6.44·22-s − 6.50·23-s − 7.58·24-s − 3.88·26-s − 27-s + 25.6·28-s − 8.17·29-s − 5.69·31-s + 11.4·32-s − 2.45·33-s + 10.8·34-s + ⋯
L(s)  = 1  + 1.85·2-s − 0.577·3-s + 2.44·4-s − 1.07·6-s + 1.98·7-s + 2.68·8-s + 0.333·9-s + 0.739·11-s − 1.41·12-s − 0.410·13-s + 3.68·14-s + 2.53·16-s + 1.00·17-s + 0.618·18-s + 0.975·19-s − 1.14·21-s + 1.37·22-s − 1.35·23-s − 1.54·24-s − 0.761·26-s − 0.192·27-s + 4.85·28-s − 1.51·29-s − 1.02·31-s + 2.02·32-s − 0.427·33-s + 1.86·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $1$
Analytic conductor: \(28.1472\)
Root analytic conductor: \(5.30539\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3525} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3525,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.880491999\)
\(L(\frac12)\) \(\approx\) \(6.880491999\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 \)
47 \( 1 - T \)
good2 \( 1 - 2.62T + 2T^{2} \)
7 \( 1 - 5.24T + 7T^{2} \)
11 \( 1 - 2.45T + 11T^{2} \)
13 \( 1 + 1.47T + 13T^{2} \)
17 \( 1 - 4.14T + 17T^{2} \)
19 \( 1 - 4.25T + 19T^{2} \)
23 \( 1 + 6.50T + 23T^{2} \)
29 \( 1 + 8.17T + 29T^{2} \)
31 \( 1 + 5.69T + 31T^{2} \)
37 \( 1 + 1.84T + 37T^{2} \)
41 \( 1 + 9.39T + 41T^{2} \)
43 \( 1 + 4.34T + 43T^{2} \)
53 \( 1 + 6.95T + 53T^{2} \)
59 \( 1 + 7.70T + 59T^{2} \)
61 \( 1 - 12.0T + 61T^{2} \)
67 \( 1 + 5.84T + 67T^{2} \)
71 \( 1 + 2.81T + 71T^{2} \)
73 \( 1 - 5.10T + 73T^{2} \)
79 \( 1 - 5.56T + 79T^{2} \)
83 \( 1 - 15.4T + 83T^{2} \)
89 \( 1 + 3.59T + 89T^{2} \)
97 \( 1 + 3.34T + 97T^{2} \)
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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

−8.121648579469193662169078973208, −7.57250998013651858087627236050, −6.93850494295070687962288054328, −5.90419672080218310291447767700, −5.33686687797606554490130795291, −4.93715835305665804462888997234, −4.05886922389827649997521666022, −3.44559458396148237016352537173, −1.99102599174699227592715415993, −1.49850531431172375957438673823, 1.49850531431172375957438673823, 1.99102599174699227592715415993, 3.44559458396148237016352537173, 4.05886922389827649997521666022, 4.93715835305665804462888997234, 5.33686687797606554490130795291, 5.90419672080218310291447767700, 6.93850494295070687962288054328, 7.57250998013651858087627236050, 8.121648579469193662169078973208

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