Properties

Label 2-6525-1.1-c1-0-160
Degree $2$
Conductor $6525$
Sign $1$
Analytic cond. $52.1023$
Root an. cond. $7.21819$
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  + 2.67·2-s + 5.15·4-s + 4.15·7-s + 8.44·8-s − 2.80·11-s − 1.35·13-s + 11.1·14-s + 12.2·16-s − 7.11·17-s + 3.76·19-s − 7.50·22-s + 4.80·23-s − 3.61·26-s + 21.4·28-s − 29-s + 0.231·31-s + 15.9·32-s − 19.0·34-s + 5.50·37-s + 10.0·38-s + 6.96·41-s + 3.19·43-s − 14.4·44-s + 12.8·46-s + 6.41·47-s + 10.2·49-s − 6.96·52-s + ⋯
L(s)  = 1  + 1.89·2-s + 2.57·4-s + 1.57·7-s + 2.98·8-s − 0.846·11-s − 0.374·13-s + 2.97·14-s + 3.06·16-s − 1.72·17-s + 0.864·19-s − 1.60·22-s + 1.00·23-s − 0.708·26-s + 4.05·28-s − 0.185·29-s + 0.0415·31-s + 2.81·32-s − 3.26·34-s + 0.905·37-s + 1.63·38-s + 1.08·41-s + 0.487·43-s − 2.18·44-s + 1.89·46-s + 0.936·47-s + 1.46·49-s − 0.965·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6525\)    =    \(3^{2} \cdot 5^{2} \cdot 29\)
Sign: $1$
Analytic conductor: \(52.1023\)
Root analytic conductor: \(7.21819\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6525,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(8.575445554\)
\(L(\frac12)\) \(\approx\) \(8.575445554\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
29 \( 1 + T \)
good2 \( 1 - 2.67T + 2T^{2} \)
7 \( 1 - 4.15T + 7T^{2} \)
11 \( 1 + 2.80T + 11T^{2} \)
13 \( 1 + 1.35T + 13T^{2} \)
17 \( 1 + 7.11T + 17T^{2} \)
19 \( 1 - 3.76T + 19T^{2} \)
23 \( 1 - 4.80T + 23T^{2} \)
31 \( 1 - 0.231T + 31T^{2} \)
37 \( 1 - 5.50T + 37T^{2} \)
41 \( 1 - 6.96T + 41T^{2} \)
43 \( 1 - 3.19T + 43T^{2} \)
47 \( 1 - 6.41T + 47T^{2} \)
53 \( 1 - 6.96T + 53T^{2} \)
59 \( 1 - 2.57T + 59T^{2} \)
61 \( 1 - 5.35T + 61T^{2} \)
67 \( 1 - 3.19T + 67T^{2} \)
71 \( 1 + 11.3T + 71T^{2} \)
73 \( 1 - 11.2T + 73T^{2} \)
79 \( 1 + 4.73T + 79T^{2} \)
83 \( 1 - 2.54T + 83T^{2} \)
89 \( 1 + 14.3T + 89T^{2} \)
97 \( 1 - 1.53T + 97T^{2} \)
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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

−7.61058891212953259106714660750, −7.24420275807121184673731034849, −6.42036244309091533270827408393, −5.44563566757214236472434662713, −5.17451432943449851298162314317, −4.44288326903754135469086145546, −3.96655781205113843168960706047, −2.60806272978240421811276511793, −2.41828493488851680615285392613, −1.24716698167141695204187473836, 1.24716698167141695204187473836, 2.41828493488851680615285392613, 2.60806272978240421811276511793, 3.96655781205113843168960706047, 4.44288326903754135469086145546, 5.17451432943449851298162314317, 5.44563566757214236472434662713, 6.42036244309091533270827408393, 7.24420275807121184673731034849, 7.61058891212953259106714660750

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