Properties

Label 2-3525-1.1-c1-0-134
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.16·2-s + 3-s − 0.641·4-s + 1.16·6-s + 4.15·7-s − 3.07·8-s + 9-s − 3.81·11-s − 0.641·12-s − 4.61·13-s + 4.84·14-s − 2.30·16-s − 1.23·17-s + 1.16·18-s − 2.53·19-s + 4.15·21-s − 4.44·22-s − 2.58·23-s − 3.07·24-s − 5.37·26-s + 27-s − 2.66·28-s − 5.43·29-s − 6.09·31-s + 3.47·32-s − 3.81·33-s − 1.43·34-s + ⋯
L(s)  = 1  + 0.824·2-s + 0.577·3-s − 0.320·4-s + 0.475·6-s + 1.57·7-s − 1.08·8-s + 0.333·9-s − 1.14·11-s − 0.185·12-s − 1.27·13-s + 1.29·14-s − 0.576·16-s − 0.298·17-s + 0.274·18-s − 0.580·19-s + 0.907·21-s − 0.947·22-s − 0.538·23-s − 0.628·24-s − 1.05·26-s + 0.192·27-s − 0.504·28-s − 1.00·29-s − 1.09·31-s + 0.613·32-s − 0.663·33-s − 0.245·34-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3525,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.16T + 2T^{2} \)
7 \( 1 - 4.15T + 7T^{2} \)
11 \( 1 + 3.81T + 11T^{2} \)
13 \( 1 + 4.61T + 13T^{2} \)
17 \( 1 + 1.23T + 17T^{2} \)
19 \( 1 + 2.53T + 19T^{2} \)
23 \( 1 + 2.58T + 23T^{2} \)
29 \( 1 + 5.43T + 29T^{2} \)
31 \( 1 + 6.09T + 31T^{2} \)
37 \( 1 + 1.49T + 37T^{2} \)
41 \( 1 - 5.04T + 41T^{2} \)
43 \( 1 + 12.3T + 43T^{2} \)
53 \( 1 + 6.27T + 53T^{2} \)
59 \( 1 - 2.67T + 59T^{2} \)
61 \( 1 - 12.7T + 61T^{2} \)
67 \( 1 + 11.3T + 67T^{2} \)
71 \( 1 + 3.49T + 71T^{2} \)
73 \( 1 - 14.2T + 73T^{2} \)
79 \( 1 - 5.30T + 79T^{2} \)
83 \( 1 - 5.94T + 83T^{2} \)
89 \( 1 + 1.23T + 89T^{2} \)
97 \( 1 - 14.5T + 97T^{2} \)
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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

−8.037510487700887798863104858967, −7.72534252874570079683155420063, −6.72697317601184868299426133159, −5.47995850026897819831781817548, −5.09683107351456964050240056761, −4.45684752072374233503678924435, −3.64238323373897045455764211477, −2.53209268819223060378467739883, −1.88836821143196245933272793949, 0, 1.88836821143196245933272793949, 2.53209268819223060378467739883, 3.64238323373897045455764211477, 4.45684752072374233503678924435, 5.09683107351456964050240056761, 5.47995850026897819831781817548, 6.72697317601184868299426133159, 7.72534252874570079683155420063, 8.037510487700887798863104858967

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