Properties

Label 2-3525-1.1-c1-0-116
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  + 2.74·2-s + 3-s + 5.52·4-s + 2.74·6-s + 1.18·7-s + 9.65·8-s + 9-s − 3.47·11-s + 5.52·12-s − 5.63·13-s + 3.24·14-s + 15.4·16-s + 4.27·17-s + 2.74·18-s + 8.18·19-s + 1.18·21-s − 9.53·22-s − 6.05·23-s + 9.65·24-s − 15.4·26-s + 27-s + 6.52·28-s + 4.15·29-s − 2.98·31-s + 23.0·32-s − 3.47·33-s + 11.7·34-s + ⋯
L(s)  = 1  + 1.93·2-s + 0.577·3-s + 2.76·4-s + 1.11·6-s + 0.446·7-s + 3.41·8-s + 0.333·9-s − 1.04·11-s + 1.59·12-s − 1.56·13-s + 0.866·14-s + 3.85·16-s + 1.03·17-s + 0.646·18-s + 1.87·19-s + 0.258·21-s − 2.03·22-s − 1.26·23-s + 1.97·24-s − 3.02·26-s + 0.192·27-s + 1.23·28-s + 0.772·29-s − 0.535·31-s + 4.06·32-s − 0.605·33-s + 2.01·34-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: \(0\)
Selberg data: \((2,\ 3525,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(8.615749512\)
\(L(\frac12)\) \(\approx\) \(8.615749512\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.74T + 2T^{2} \)
7 \( 1 - 1.18T + 7T^{2} \)
11 \( 1 + 3.47T + 11T^{2} \)
13 \( 1 + 5.63T + 13T^{2} \)
17 \( 1 - 4.27T + 17T^{2} \)
19 \( 1 - 8.18T + 19T^{2} \)
23 \( 1 + 6.05T + 23T^{2} \)
29 \( 1 - 4.15T + 29T^{2} \)
31 \( 1 + 2.98T + 31T^{2} \)
37 \( 1 - 6.90T + 37T^{2} \)
41 \( 1 - 7.34T + 41T^{2} \)
43 \( 1 - 2.62T + 43T^{2} \)
53 \( 1 + 1.77T + 53T^{2} \)
59 \( 1 - 2.60T + 59T^{2} \)
61 \( 1 + 9.07T + 61T^{2} \)
67 \( 1 + 12.8T + 67T^{2} \)
71 \( 1 - 1.83T + 71T^{2} \)
73 \( 1 + 1.43T + 73T^{2} \)
79 \( 1 + 2.99T + 79T^{2} \)
83 \( 1 + 6.78T + 83T^{2} \)
89 \( 1 + 11.7T + 89T^{2} \)
97 \( 1 + 14.9T + 97T^{2} \)
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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

−7.944720216932066534486198557645, −7.68427063929009075256858205156, −7.14826139112858339170244956859, −5.94770619536428723205465562540, −5.37975745968869453970772624928, −4.76522911794622169781507430315, −4.02724309840002413442127322319, −2.90246601118922328172469845467, −2.67835022438671089103787789059, −1.49458112756450719028670485718, 1.49458112756450719028670485718, 2.67835022438671089103787789059, 2.90246601118922328172469845467, 4.02724309840002413442127322319, 4.76522911794622169781507430315, 5.37975745968869453970772624928, 5.94770619536428723205465562540, 7.14826139112858339170244956859, 7.68427063929009075256858205156, 7.944720216932066534486198557645

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